course notes 4.5: matrix inverses 4.6: determinants...
TRANSCRIPT
![Page 1: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/1.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Outline
Week 8: Inverses and determinants
Course Notes: 4.5, 4.6
Goals: Be able to calculate a matrix’s inverse;understand the relationship between the invertibility of a matrixand the solutions of associated linear systems;calculate the determinant of a square matrix of any size, and learnsome tricks to make the computation more efficient.
![Page 2: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/2.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Calculate: 1 0 00 1 00 0 1
1 2 34 5 67 8 9
Calculate: [1 23 4
] [1 00 1
]
![Page 3: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/3.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Identity Matrix
I =
1 0 0 0 0 0 0 00 1 0 0 · · · 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 0
.... . .
...0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 · · · 0 0 1 00 0 0 0 0 0 0 1
The identity matrix, I , is a square matrix with 1s along its maindiagonal, and 0s everywhere else.
For any matrix A that can be multiplied with I , AI = IA = A.
![Page 4: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/4.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0
There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 5: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/5.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0
There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 6: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/6.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0
There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 7: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/7.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0
There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 8: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/8.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0
There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 9: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/9.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0
There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 10: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/10.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 11: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/11.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 12: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/12.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
(a + 5)x = 7x
Divide both sides by x ***as long as*** x 6= 0There are some numbers we can’t divide by.
(a + 5)x
x= 7
x
x
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
(a + 5)(1) = 7(1)
1 is the multiplicative identity. If you multiply it by a number, thatnumber doesn’t change.
(a + 5) = 7
![Page 13: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/13.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
To replicate “division” in matrices, we want to find a matrix A(called A−1) with the property that AA−1 = I , the identity matrix.
For example, 4× 0.25 = 1, so dividing by 4 is the same asmultiplying by 0.25.
0.1× 10 = 1, so dividing by 0.1 is the same as multiplying by 10.
We can’t divide by 0 because there is NO number x such that0× x = 1.
There are MANY matrices A such that AB 6= I no matter whatmatrix B we try.
![Page 14: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/14.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
To replicate “division” in matrices, we want to find a matrix A(called A−1) with the property that AA−1 = I , the identity matrix.
For example, 4× 0.25 = 1, so dividing by 4 is the same asmultiplying by 0.25.
0.1× 10 = 1, so dividing by 0.1 is the same as multiplying by 10.
We can’t divide by 0 because there is NO number x such that0× x = 1.
There are MANY matrices A such that AB 6= I no matter whatmatrix B we try.
![Page 15: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/15.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
To replicate “division” in matrices, we want to find a matrix A(called A−1) with the property that AA−1 = I , the identity matrix.
For example, 4× 0.25 = 1, so dividing by 4 is the same asmultiplying by 0.25.
0.1× 10 = 1, so dividing by 0.1 is the same as multiplying by 10.
We can’t divide by 0 because there is NO number x such that0× x = 1.
There are MANY matrices A such that AB 6= I no matter whatmatrix B we try.
![Page 16: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/16.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.
To replicate “division” in matrices, we want to find a matrix A(called A−1) with the property that AA−1 = I , the identity matrix.
For example, 4× 0.25 = 1, so dividing by 4 is the same asmultiplying by 0.25.
0.1× 10 = 1, so dividing by 0.1 is the same as multiplying by 10.
We can’t divide by 0 because there is NO number x such that0× x = 1.
There are MANY matrices A such that AB 6= I no matter whatmatrix B we try.
![Page 17: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/17.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.To replicate “division” in matrices, we want to find a matrix A(called A−1) with the property that AA−1 = I , the identity matrix.
For example, 4× 0.25 = 1, so dividing by 4 is the same asmultiplying by 0.25.
0.1× 10 = 1, so dividing by 0.1 is the same as multiplying by 10.
We can’t divide by 0 because there is NO number x such that0× x = 1.
There are MANY matrices A such that AB 6= I no matter whatmatrix B we try.
![Page 18: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/18.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
What is Division?
To divide by x , we multiply by a special number (in this case, 1/x)that has the following property: x(1/x) gives the multiplicativeidentity.To replicate “division” in matrices, we want to find a matrix A(called A−1) with the property that AA−1 = I , the identity matrix.
For example, 4× 0.25 = 1, so dividing by 4 is the same asmultiplying by 0.25.
0.1× 10 = 1, so dividing by 0.1 is the same as multiplying by 10.
We can’t divide by 0 because there is NO number x such that0× x = 1.There are MANY matrices A such that AB 6= I no matter whatmatrix B we try.
![Page 19: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/19.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Matrix Inverses: The Closest we can Get to Division
Linear System Setup: x + 2y + 3z = 104x + 5y + 6z = 207x + 8y + 9z = 30
A =
1 2 34 5 67 8 9
x =
xyz
b =
102030
Ax = b
Solve for x .
Can’t we just divide by A?Wanted: a matrix A−1 with the property A−1A = I , identity matrix.Then: A−1Ax = A−1b, so x = A−1b
![Page 20: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/20.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Matrix Inverses: The Closest we can Get to Division
Linear System Setup: x + 2y + 3z = 104x + 5y + 6z = 207x + 8y + 9z = 30
A =
1 2 34 5 67 8 9
x =
xyz
b =
102030
Ax = b
Solve for x .
Can’t we just divide by A?Wanted: a matrix A−1 with the property A−1A = I , identity matrix.Then: A−1Ax = A−1b, so x = A−1b
![Page 21: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/21.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Matrix Inverses: The Closest we can Get to Division
Linear System Setup: x + 2y + 3z = 104x + 5y + 6z = 207x + 8y + 9z = 30
A =
1 2 34 5 67 8 9
x =
xyz
b =
102030
Ax = b
Solve for x .
Can’t we just divide by A?Wanted: a matrix A−1 with the property A−1A = I , identity matrix.Then: A−1Ax = A−1b, so x = A−1b
![Page 22: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/22.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Matrix Inverses: The Closest we can Get to Division
Linear System Setup: x + 2y + 3z = 104x + 5y + 6z = 207x + 8y + 9z = 30
A =
1 2 34 5 67 8 9
x =
xyz
b =
102030
Ax = b
Solve for x .
Can’t we just divide by A?
Wanted: a matrix A−1 with the property A−1A = I , identity matrix.Then: A−1Ax = A−1b, so x = A−1b
![Page 23: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/23.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Matrix Inverses: The Closest we can Get to Division
Linear System Setup: x + 2y + 3z = 104x + 5y + 6z = 207x + 8y + 9z = 30
A =
1 2 34 5 67 8 9
x =
xyz
b =
102030
Ax = b
Solve for x .
Can’t we just divide by A?Wanted: a matrix A−1 with the property A−1A = I , identity matrix.
Then: A−1Ax = A−1b, so x = A−1b
![Page 24: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/24.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Matrix Inverses: The Closest we can Get to Division
Linear System Setup: x + 2y + 3z = 104x + 5y + 6z = 207x + 8y + 9z = 30
A =
1 2 34 5 67 8 9
x =
xyz
b =
102030
Ax = b
Solve for x .
Can’t we just divide by A?Wanted: a matrix A−1 with the property A−1A = I , identity matrix.Then: A−1Ax = A−1b, so x = A−1b
![Page 25: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/25.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Definition
A matrix A−1 is the inverse of a square matrix A if A−1A = I ,where I is the identity matrix. In this case, alsoa AA−1 = I .
awe won’t prove this bit
What do you think the inverse of the following matrix should be?[cos θ − sin θsin θ cos θ
]
What do you think the inverse of the following matrix should be?[cos θ sin θsin θ − cos θ
]
![Page 26: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/26.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Definition
A matrix A−1 is the inverse of a square matrix A if A−1A = I ,where I is the identity matrix. In this case, alsoa AA−1 = I .
awe won’t prove this bit
What do you think the inverse of the following matrix should be?[cos θ − sin θsin θ cos θ
]
What do you think the inverse of the following matrix should be?[cos θ sin θsin θ − cos θ
]
![Page 27: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/27.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Definition
A matrix A−1 is the inverse of a square matrix A if A−1A = I ,where I is the identity matrix. In this case, alsoa AA−1 = I .
awe won’t prove this bit
What do you think the inverse of the following matrix should be?[cos θ − sin θsin θ cos θ
]
What do you think the inverse of the following matrix should be?[cos θ sin θsin θ − cos θ
]
![Page 28: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/28.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Check:
[cos θ − sin θsin θ cos θ
] [cos(−θ) − sin(−θ)sin(−θ) cos(−θ)
]=
[cos θ − sin θsin θ cos θ
] [cos θ sin θ− sin θ cos θ
]=
[cos2 θ + sin2 θ 0
0 sin2 θ + cos2 θ
]=
[1 00 1
]
[cos θ sin θsin θ − cos θ
] [cos θ sin θsin θ − cos θ
]=
[cos2 θ + sin2 θ 0
0 sin2 θ + cos2 θ
]=
[1 00 1
]
![Page 29: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/29.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Check:
[cos θ − sin θsin θ cos θ
] [cos(−θ) − sin(−θ)sin(−θ) cos(−θ)
]=
[cos θ − sin θsin θ cos θ
] [cos θ sin θ− sin θ cos θ
]=
[cos2 θ + sin2 θ 0
0 sin2 θ + cos2 θ
]=
[1 00 1
]
[cos θ sin θsin θ − cos θ
] [cos θ sin θsin θ − cos θ
]=
[cos2 θ + sin2 θ 0
0 sin2 θ + cos2 θ
]=
[1 00 1
]
![Page 30: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/30.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Check:
[cos θ − sin θsin θ cos θ
] [cos(−θ) − sin(−θ)sin(−θ) cos(−θ)
]=
[cos θ − sin θsin θ cos θ
] [cos θ sin θ− sin θ cos θ
]=
[cos2 θ + sin2 θ 0
0 sin2 θ + cos2 θ
]=
[1 00 1
]
[cos θ sin θsin θ − cos θ
] [cos θ sin θsin θ − cos θ
]=
[cos2 θ + sin2 θ 0
0 sin2 θ + cos2 θ
]=
[1 00 1
]
![Page 31: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/31.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Existence of Matrix Inverses
Definition
A matrix A−1 is the inverse of a square matrix A if
A−1A = I
where I is the identity matrix.
Find the inverses of the following matrices:
A =
[2 10 1
]B =
[1 03 1
]C =
[0 00 0
]D =
[1 11 1
]
A−1 =
[12 − 1
20 1
]B−1 =
[1 0−3 1
]If Ax = b and A−1 exists, then x = A−1bIf A−1 exists, then Ax = b has a unique solution.
![Page 32: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/32.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Existence of Matrix Inverses
Definition
A matrix A−1 is the inverse of a square matrix A if
A−1A = I
where I is the identity matrix.
Find the inverses of the following matrices:
A =
[2 10 1
]B =
[1 03 1
]C =
[0 00 0
]D =
[1 11 1
]
A−1 =
[12 − 1
20 1
]B−1 =
[1 0−3 1
]If Ax = b and A−1 exists, then x = A−1bIf A−1 exists, then Ax = b has a unique solution.
![Page 33: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/33.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Existence of Matrix Inverses
Definition
A matrix A−1 is the inverse of a square matrix A if
A−1A = I
where I is the identity matrix.
Find the inverses of the following matrices:
A =
[2 10 1
]B =
[1 03 1
]C =
[0 00 0
]D =
[1 11 1
]
A−1 =
[12 − 1
20 1
]
B−1 =
[1 0−3 1
]If Ax = b and A−1 exists, then x = A−1bIf A−1 exists, then Ax = b has a unique solution.
![Page 34: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/34.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Existence of Matrix Inverses
Definition
A matrix A−1 is the inverse of a square matrix A if
A−1A = I
where I is the identity matrix.
Find the inverses of the following matrices:
A =
[2 10 1
]B =
[1 03 1
]C =
[0 00 0
]D =
[1 11 1
]
A−1 =
[12 − 1
20 1
]B−1 =
[1 0−3 1
]
If Ax = b and A−1 exists, then x = A−1bIf A−1 exists, then Ax = b has a unique solution.
![Page 35: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/35.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Existence of Matrix Inverses
Definition
A matrix A−1 is the inverse of a square matrix A if
A−1A = I
where I is the identity matrix.
Find the inverses of the following matrices:
A =
[2 10 1
]B =
[1 03 1
]C =
[0 00 0
]D =
[1 11 1
]
A−1 =
[12 − 1
20 1
]B−1 =
[1 0−3 1
]If Ax = b and A−1 exists, then x = A−1b
If A−1 exists, then Ax = b has a unique solution.
![Page 36: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/36.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Existence of Matrix Inverses
Definition
A matrix A−1 is the inverse of a square matrix A if
A−1A = I
where I is the identity matrix.
Find the inverses of the following matrices:
A =
[2 10 1
]B =
[1 03 1
]C =
[0 00 0
]D =
[1 11 1
]
A−1 =
[12 − 1
20 1
]B−1 =
[1 0−3 1
]If Ax = b and A−1 exists, then x = A−1bIf A−1 exists, then Ax = b has a unique solution.
![Page 37: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/37.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If an Inverse Exists....
Theorem
If an n-by-n matrix A has an inverse A−1, then for any b in Rn,
Ax = b
has precisely one solution, and that solution is
x = A−1b.
So, if Ax = b has no solutions:A is not invertible.
If Ax = b has infinitely many solutions:A is not invertible.
![Page 38: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/38.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If an Inverse Exists....
Theorem
If an n-by-n matrix A has an inverse A−1, then for any b in Rn,
Ax = b
has precisely one solution, and that solution is
x = A−1b.
So, if Ax = b has no solutions:
A is not invertible.
If Ax = b has infinitely many solutions:A is not invertible.
![Page 39: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/39.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If an Inverse Exists....
Theorem
If an n-by-n matrix A has an inverse A−1, then for any b in Rn,
Ax = b
has precisely one solution, and that solution is
x = A−1b.
So, if Ax = b has no solutions:A is not invertible.
If Ax = b has infinitely many solutions:A is not invertible.
![Page 40: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/40.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If an Inverse Exists....
Theorem
If an n-by-n matrix A has an inverse A−1, then for any b in Rn,
Ax = b
has precisely one solution, and that solution is
x = A−1b.
So, if Ax = b has no solutions:A is not invertible.
If Ax = b has infinitely many solutions:
A is not invertible.
![Page 41: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/41.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If an Inverse Exists....
Theorem
If an n-by-n matrix A has an inverse A−1, then for any b in Rn,
Ax = b
has precisely one solution, and that solution is
x = A−1b.
So, if Ax = b has no solutions:A is not invertible.
If Ax = b has infinitely many solutions:A is not invertible.
![Page 42: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/42.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.
statements 1-4
invertible
emp
ty
invertible
???
Are the statementsequivalent to Abeing invertible?
![Page 43: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/43.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.
statements 1-4
invertible
emp
ty
invertible
???
Are the statementsequivalent to Abeing invertible?
![Page 44: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/44.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.
statements 1-4invertible
emp
ty
invertible
???
Are the statementsequivalent to Abeing invertible?
![Page 45: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/45.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.
statements 1-4invertible
emp
ty
invertible
???
Are the statementsequivalent to Abeing invertible?
![Page 46: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/46.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.
statements 1-4
invertible
emp
ty
invertible
???
Are the statementsequivalent to Abeing invertible?
![Page 47: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/47.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.
statements 1-4
invertible
emp
ty
invertible
???
Are the statementsequivalent to Abeing invertible?
![Page 48: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/48.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
Rn Rn
x1 b1
x2 b2
x3 b3
x4 b4
T (x1) = Ax1 = b1
T (x2) = Ax2 = b2
T (x3) = Ax3 = b3
T (x4) = Ax4 = b4
T−1(b1) = x1
T−1(b2) = x2
T−1(b3) = x3
T−1(b4) = x4
If T−1 is a linear transformation, then we can find a matrix B such that
T−1(b) = Bb
for every b. Then: x = Bb = B(Ax) = (BA)x, so BA = I
![Page 49: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/49.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
Rn Rn
x1 b1
x2 b2
x3 b3
x4 b4
T (x1) = Ax1 = b1
T (x2) = Ax2 = b2
T (x3) = Ax3 = b3
T (x4) = Ax4 = b4
T−1(b1) = x1
T−1(b2) = x2
T−1(b3) = x3
T−1(b4) = x4
If T−1 is a linear transformation, then we can find a matrix B such that
T−1(b) = Bb
for every b. Then: x = Bb = B(Ax) = (BA)x, so BA = I
![Page 50: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/50.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
Rn Rn
x1 b1
x2 b2
x3 b3
x4 b4
T (x1) = Ax1 = b1
T (x2) = Ax2 = b2
T (x3) = Ax3 = b3
T (x4) = Ax4 = b4
T−1(b1) = x1
T−1(b2) = x2
T−1(b3) = x3
T−1(b4) = x4
If T−1 is a linear transformation, then we can find a matrix B such that
T−1(b) = Bb
for every b. Then: x = Bb = B(Ax) = (BA)x, so BA = I
![Page 51: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/51.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
Rn Rn
x1 b1
x2 b2
x3 b3
x4 b4
T (x1) = Ax1 = b1
T (x2) = Ax2 = b2
T (x3) = Ax3 = b3
T (x4) = Ax4 = b4
T−1(b1) = x1
T−1(b2) = x2
T−1(b3) = x3
T−1(b4) = x4
If T−1 is a linear transformation, then we can find a matrix B such that
T−1(b) = Bb
for every b. Then: x = Bb = B(Ax) = (BA)x, so BA = I
![Page 52: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/52.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
Rn Rn
x1 b1
x2 b2
x3 b3
x4 b4
T (x1) = Ax1 = b1
T (x2) = Ax2 = b2
T (x3) = Ax3 = b3
T (x4) = Ax4 = b4
T−1(b1) = x1
T−1(b2) = x2
T−1(b3) = x3
T−1(b4) = x4
If T−1 is a linear transformation, then we can find a matrix B such that
T−1(b) = Bb
for every b.
Then: x = Bb = B(Ax) = (BA)x, so BA = I
![Page 53: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/53.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
Rn Rn
x1 b1
x2 b2
x3 b3
x4 b4
T (x1) = Ax1 = b1
T (x2) = Ax2 = b2
T (x3) = Ax3 = b3
T (x4) = Ax4 = b4
T−1(b1) = x1
T−1(b2) = x2
T−1(b3) = x3
T−1(b4) = x4
If T−1 is a linear transformation, then we can find a matrix B such that
T−1(b) = Bb
for every b. Then: x = Bb = B(Ax) = (BA)x, so BA = I
![Page 54: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/54.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
Need to show: T−1 is a linear transformation.
• Fix A.
• Given b, we can solve Ax = b for x.
• So, given b, we find x.
• This is a transformation: T−1(b) = x. That is, given input b, theoutput x is the vector we multiply A by to get b.
• T−1 is linear:
- Let T−1(b1) = x1 and T−1(b2) = x2.- Note A(x1 + x2) = Ax1 + Ax2 = b1 + b2.
So, T−1(b1 + b2) = x1 + x2 = T−1(b1) + T−1(b2).So, T−1 preserves addition.
- Note A(sx1) = sA(x1) = sb1, so T−1(sb1) = sx1 = sT−1(b1).So, T−1 preserves scalar multiplication.
• Since T−1 is a linear transformation from one collection of vectorsto another, there exists some matrix B such that T−1(b) = Bb.
• Consider T−1(Ax). Note T−1(Ax) = x for every x in Rn, soB(Ax) = x for every x. Therefore, BA = I , so B = A−1.
![Page 55: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/55.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If Ax = b has a unique solution for every b, is A invertible?
• Fix A.
• Given b, we can solve Ax = b for x.
• So, given b, we find x.
• This is a transformation: T−1(b) = x. That is, given input b, theoutput x is the vector we multiply A by to get b.
• T−1 is linear:
- Let T−1(b1) = x1 and T−1(b2) = x2.- Note A(x1 + x2) = Ax1 + Ax2 = b1 + b2.
So, T−1(b1 + b2) = x1 + x2 = T−1(b1) + T−1(b2).So, T−1 preserves addition.
- Note A(sx1) = sA(x1) = sb1, so T−1(sb1) = sx1 = sT−1(b1).So, T−1 preserves scalar multiplication.
• Since T−1 is a linear transformation from one collection of vectorsto another, there exists some matrix B such that T−1(b) = Bb.
• Consider T−1(Ax). Note T−1(Ax) = x for every x in Rn, soB(Ax) = x for every x. Therefore, BA = I , so B = A−1.
![Page 56: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/56.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.And now we’ve shown that if the statements hold, then A is invertible
statements 1-4
invertible
emp
ty
invertible
![Page 57: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/57.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.And now we’ve shown that if the statements hold, then A is invertible
statements 1-4
invertible
emp
ty
invertible
![Page 58: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/58.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
By previous theorem, if A is invertible, all these statements hold.And now we’ve shown that if the statements hold, then A is invertible
statements 1-4
invertible
emp
ty
invertible
![Page 59: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/59.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
5) A is invertible
statements 1-4
invertible
emp
ty
invertible
![Page 60: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/60.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Theorem:A is invertible if and only if Ax = b has exactly one solution forevery b .
Suppose A is a matrix with the following reduced form.Is A invertible?
1 0 30 1 20 0 0
no
1 0 00 1 00 0 1
yes
1 0 00 0 10 0 0
no
![Page 61: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/61.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Theorem:A is invertible if and only if Ax = b has exactly one solution forevery b .
Suppose A is a matrix with the following reduced form.Is A invertible?
1 0 30 1 20 0 0
no
1 0 00 1 00 0 1
yes
1 0 00 0 10 0 0
no
![Page 62: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/62.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Theorem:A is invertible if and only if Ax = b has exactly one solution forevery b .
Suppose A is a matrix with the following reduced form.Is A invertible?
1 0 30 1 20 0 0
no
1 0 00 1 00 0 1
yes
1 0 00 0 10 0 0
no
![Page 63: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/63.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Theorem:A is invertible if and only if Ax = b has exactly one solution forevery b .
Suppose A is a matrix with the following reduced form.Is A invertible?
1 0 30 1 20 0 0
no
1 0 00 1 00 0 1
yes
1 0 00 0 10 0 0
no
![Page 64: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/64.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Theorem:A is invertible if and only if Ax = b has exactly one solution forevery b .
Suppose A is a matrix with the following reduced form.Is A invertible?
1 0 30 1 20 0 0
no
1 0 00 1 00 0 1
yes
1 0 00 0 10 0 0
no
![Page 65: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/65.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.
Row 1 → 2(Row 1)
2 0 00 1 00 0 1
a b cd e fg h i
=
Row 1 → (Row 1 + Row 2)
1 1 00 1 00 0 1
a b cd e fg h i
=
Row 1 ↔ Row 3
0 0 10 1 01 0 0
a b cd e fg h i
=
![Page 66: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/66.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.
Row 1 → 2(Row 1)
2 0 00 1 00 0 1
a b cd e fg h i
=
Row 1 → (Row 1 + Row 2)
1 1 00 1 00 0 1
a b cd e fg h i
=
Row 1 ↔ Row 3
0 0 10 1 01 0 0
a b cd e fg h i
=
![Page 67: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/67.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.Row 1 → 2(Row 1) 2 0 0
0 1 00 0 1
a b cd e fg h i
=
Row 1 → (Row 1 + Row 2)
1 1 00 1 00 0 1
a b cd e fg h i
=
Row 1 ↔ Row 3
0 0 10 1 01 0 0
a b cd e fg h i
=
![Page 68: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/68.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.Row 1 → 2(Row 1) 2 0 0
0 1 00 0 1
a b cd e fg h i
=
Row 1 → (Row 1 + Row 2)1 1 00 1 00 0 1
a b cd e fg h i
=
Row 1 ↔ Row 3
0 0 10 1 01 0 0
a b cd e fg h i
=
![Page 69: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/69.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.Row 1 → 2(Row 1) 2 0 0
0 1 00 0 1
a b cd e fg h i
=
Row 1 → (Row 1 + Row 2)1 1 00 1 00 0 1
a b cd e fg h i
=
Row 1 ↔ Row 3 0 0 10 1 01 0 0
a b cd e fg h i
=
![Page 70: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/70.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.
A→→→ I
We row-reduce A until it becomes the identity matrix. Theoperations used are equivalent to multiplications:
[E3E2E1]A = I
So, [E3E2E1] = E is the inverse of A. But, how do we find it?
EI = E
so, we reduce I at the same time as A, using the same operations.
I →→→ [E3E2E1]I = E
![Page 71: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/71.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.
A→→→ I
We row-reduce A until it becomes the identity matrix. Theoperations used are equivalent to multiplications:
[E3E2E1]A = I
So, [E3E2E1] = E is the inverse of A. But, how do we find it?
EI = E
so, we reduce I at the same time as A, using the same operations.
I →→→ [E3E2E1]I = E
![Page 72: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/72.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.
A→→→ I
We row-reduce A until it becomes the identity matrix. Theoperations used are equivalent to multiplications:
[E3E2E1]A = I
So, [E3E2E1] = E is the inverse of A. But, how do we find it?
EI = E
so, we reduce I at the same time as A, using the same operations.
I →→→ [E3E2E1]I = E
![Page 73: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/73.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.
A→→→ I
We row-reduce A until it becomes the identity matrix. Theoperations used are equivalent to multiplications:
[E3E2E1]A = I
So, [E3E2E1] = E is the inverse of A. But, how do we find it?
EI = E
so, we reduce I at the same time as A, using the same operations.
I →→→ [E3E2E1]I = E
![Page 74: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/74.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
An observation that will help compute inverses
Elementary row operations are equivalent to matrix multiplication.
A→→→ I
We row-reduce A until it becomes the identity matrix. Theoperations used are equivalent to multiplications:
[E3E2E1]A = I
So, [E3E2E1] = E is the inverse of A. But, how do we find it?
EI = E
so, we reduce I at the same time as A, using the same operations.
I →→→ [E3E2E1]I = E
![Page 75: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/75.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Conveniently Computing the Inverse (when it exists)
[A I
] reduce−−−−→[I A−1
]
[2 1 1 00 1 0 1
]︸ ︷︷ ︸
[A|I ]
R1−R2−−−−→[
2 0 1 −10 1 0 1
]12R1−−→[
1 0 12−12
0 1 0 1
]︸ ︷︷ ︸
[I |A−1]
Calculate the inverse of A =
1 0 32 1 62 0 7
A−1 =
7 0 −3−2 1 0−2 0 1
Calculate the inverse of B =
[1 11 1
]
![Page 76: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/76.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Conveniently Computing the Inverse (when it exists)
[A I
] reduce−−−−→[I A−1
]
[2 1 1 00 1 0 1
]︸ ︷︷ ︸
[A|I ]
R1−R2−−−−→[
2 0 1 −10 1 0 1
]12R1−−→[
1 0 12−12
0 1 0 1
]︸ ︷︷ ︸
[I |A−1]
Calculate the inverse of A =
1 0 32 1 62 0 7
A−1 =
7 0 −3−2 1 0−2 0 1
Calculate the inverse of B =
[1 11 1
]
![Page 77: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/77.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Conveniently Computing the Inverse (when it exists)
[A I
] reduce−−−−→[I A−1
]
[2 1 1 00 1 0 1
]︸ ︷︷ ︸
[A|I ]
R1−R2−−−−→[
2 0 1 −10 1 0 1
]
12R1−−→[
1 0 12−12
0 1 0 1
]︸ ︷︷ ︸
[I |A−1]
Calculate the inverse of A =
1 0 32 1 62 0 7
A−1 =
7 0 −3−2 1 0−2 0 1
Calculate the inverse of B =
[1 11 1
]
![Page 78: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/78.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Conveniently Computing the Inverse (when it exists)
[A I
] reduce−−−−→[I A−1
]
[2 1 1 00 1 0 1
]︸ ︷︷ ︸
[A|I ]
R1−R2−−−−→[
2 0 1 −10 1 0 1
]12R1−−→[
1 0 12−12
0 1 0 1
]︸ ︷︷ ︸
[I |A−1]
Calculate the inverse of A =
1 0 32 1 62 0 7
A−1 =
7 0 −3−2 1 0−2 0 1
Calculate the inverse of B =
[1 11 1
]
![Page 79: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/79.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Conveniently Computing the Inverse (when it exists)
[A I
] reduce−−−−→[I A−1
]
[2 1 1 00 1 0 1
]︸ ︷︷ ︸
[A|I ]
R1−R2−−−−→[
2 0 1 −10 1 0 1
]12R1−−→[
1 0 12−12
0 1 0 1
]︸ ︷︷ ︸
[I |A−1]
Calculate the inverse of A =
1 0 32 1 62 0 7
A−1 =
7 0 −3−2 1 0−2 0 1
Calculate the inverse of B =
[1 11 1
]
![Page 80: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/80.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Conveniently Computing the Inverse (when it exists)
[A I
] reduce−−−−→[I A−1
]
[2 1 1 00 1 0 1
]︸ ︷︷ ︸
[A|I ]
R1−R2−−−−→[
2 0 1 −10 1 0 1
]12R1−−→[
1 0 12−12
0 1 0 1
]︸ ︷︷ ︸
[I |A−1]
Calculate the inverse of A =
1 0 32 1 62 0 7
A−1 =
7 0 −3−2 1 0−2 0 1
Calculate the inverse of B =
[1 11 1
]
![Page 81: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/81.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Conveniently Computing the Inverse (when it exists)
[A I
] reduce−−−−→[I A−1
]
[2 1 1 00 1 0 1
]︸ ︷︷ ︸
[A|I ]
R1−R2−−−−→[
2 0 1 −10 1 0 1
]12R1−−→[
1 0 12−12
0 1 0 1
]︸ ︷︷ ︸
[I |A−1]
Calculate the inverse of A =
1 0 32 1 62 0 7
A−1 =
7 0 −3−2 1 0−2 0 1
Calculate the inverse of B =
[1 11 1
]
![Page 82: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/82.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Inverses
Suppose M = [ 2 10 1 ]. Then (as we just found) M−1 =
[1/2 −1/20 1
].
If Mx = [ 518 ], what is x ?
Fact: the equation[ 1 32 6 ] x = [ 5
18 ]
has NO solution x . The matrix [ 1 32 6 ] is not invertible.
Suppose A =[1 0 32 1 62 0 7
]and A−1 =
[ 7 0 −3−2 1 0−2 0 1
].
If BA =[1 1 11 1 11 1 1
], what is B?
Fact: the equation
B[1 0 31 0 32 0 7
]=[1 1 11 1 11 1 1
]has INFINITELY MANY solutions: B =
[a (5−a) −2a (5−a) −2a (5−a) −2
].
![Page 83: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/83.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Inverses
Suppose M = [ 2 10 1 ]. Then (as we just found) M−1 =
[1/2 −1/20 1
].
If Mx = [ 518 ], what is x ?
Fact: the equation[ 1 32 6 ] x = [ 5
18 ]
has NO solution x . The matrix [ 1 32 6 ] is not invertible.
Suppose A =[1 0 32 1 62 0 7
]and A−1 =
[ 7 0 −3−2 1 0−2 0 1
].
If BA =[1 1 11 1 11 1 1
], what is B?
Fact: the equation
B[1 0 31 0 32 0 7
]=[1 1 11 1 11 1 1
]has INFINITELY MANY solutions: B =
[a (5−a) −2a (5−a) −2a (5−a) −2
].
![Page 84: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/84.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Inverses
Suppose M = [ 2 10 1 ]. Then (as we just found) M−1 =
[1/2 −1/20 1
].
If Mx = [ 518 ], what is x ?
Fact: the equation[ 1 32 6 ] x = [ 5
18 ]
has NO solution x . The matrix [ 1 32 6 ] is not invertible.
Suppose A =[1 0 32 1 62 0 7
]and A−1 =
[ 7 0 −3−2 1 0−2 0 1
].
If BA =[1 1 11 1 11 1 1
], what is B?
Fact: the equation
B[1 0 31 0 32 0 7
]=[1 1 11 1 11 1 1
]has INFINITELY MANY solutions: B =
[a (5−a) −2a (5−a) −2a (5−a) −2
].
![Page 85: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/85.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Inverses
Suppose M = [ 2 10 1 ]. Then (as we just found) M−1 =
[1/2 −1/20 1
].
If Mx = [ 518 ], what is x ?
Fact: the equation[ 1 32 6 ] x = [ 5
18 ]
has NO solution x . The matrix [ 1 32 6 ] is not invertible.
Suppose A =[1 0 32 1 62 0 7
]and A−1 =
[ 7 0 −3−2 1 0−2 0 1
].
If BA =[1 1 11 1 11 1 1
], what is B?
Fact: the equation
B[1 0 31 0 32 0 7
]=[1 1 11 1 11 1 1
]has INFINITELY MANY solutions: B =
[a (5−a) −2a (5−a) −2a (5−a) −2
].
![Page 86: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/86.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Inverses
Suppose M = [ 2 10 1 ]. Then (as we just found) M−1 =
[1/2 −1/20 1
].
If Mx = [ 518 ], what is x ?
Fact: the equation[ 1 32 6 ] x = [ 5
18 ]
has NO solution x . The matrix [ 1 32 6 ] is not invertible.
Suppose A =[1 0 32 1 62 0 7
]and A−1 =
[ 7 0 −3−2 1 0−2 0 1
].
If BA =[1 1 11 1 11 1 1
], what is B?
Fact: the equation
B[1 0 31 0 32 0 7
]=[1 1 11 1 11 1 1
]has INFINITELY MANY solutions: B =
[a (5−a) −2a (5−a) −2a (5−a) −2
].
![Page 87: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/87.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Inverses
Suppose M = [ 2 10 1 ]. Then (as we just found) M−1 =
[1/2 −1/20 1
].
If Mx = [ 518 ], what is x ?
Fact: the equation[ 1 32 6 ] x = [ 5
18 ]
has NO solution x . The matrix [ 1 32 6 ] is not invertible.
Suppose A =[1 0 32 1 62 0 7
]and A−1 =
[ 7 0 −3−2 1 0−2 0 1
].
If BA =[1 1 11 1 11 1 1
], what is B?
Fact: the equation
B[1 0 31 0 32 0 7
]=[1 1 11 1 11 1 1
]has INFINITELY MANY solutions: B =
[a (5−a) −2a (5−a) −2a (5−a) −2
].
![Page 88: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/88.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Inverses and Products
Suppose A and B are invertible matrices, with the samedimensions. Simplify:
ABB−1A−1
Since ABB−1A−1 = I , we conclude that the inverse of AB isB−1A−1.
What is (ABC )−1?
Simplify: [(AC )−1A(AB)−1
]−1
![Page 89: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/89.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Inverses and Products
Suppose A and B are invertible matrices, with the samedimensions. Simplify:
ABB−1A−1
Since ABB−1A−1 = I , we conclude that the inverse of AB isB−1A−1.
What is (ABC )−1?
Simplify: [(AC )−1A(AB)−1
]−1
![Page 90: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/90.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Inverses and Products
Suppose A and B are invertible matrices, with the samedimensions. Simplify:
ABB−1A−1
Since ABB−1A−1 = I , we conclude that the inverse of AB isB−1A−1.
What is (ABC )−1?
Simplify: [(AC )−1A(AB)−1
]−1
![Page 91: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/91.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Inverses and Products
Suppose A and B are invertible matrices, with the samedimensions. Simplify:
ABB−1A−1
Since ABB−1A−1 = I , we conclude that the inverse of AB isB−1A−1.
What is (ABC )−1?
Simplify: [(AC )−1A(AB)−1
]−1
![Page 92: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/92.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants
Recall:
det
[a bc d
]= ad − bc
det
= a det−b det +c det
In general:
det
a1,1 a1,2 · · · a1,na2,1 a2,2 · · · a2,n
...an,1 an,2 · · · an,n
= a1,1D1,1−a1,2D1,2+a1,3D1,3 · · ·±a1,nD1,n
where Di ,j is the determinant of the matrix obtained from Aby deleting row i and column j .
![Page 93: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/93.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants
Recall:
det
[a bc d
]= ad − bc
det
a b cd e fg h i
= a det
[e fh i
]− b det
[d fg i
]+ c det
[d eg h
]
In general:
det
a1,1 a1,2 · · · a1,na2,1 a2,2 · · · a2,n
...an,1 an,2 · · · an,n
= a1,1D1,1−a1,2D1,2+a1,3D1,3 · · ·±a1,nD1,n
where Di ,j is the determinant of the matrix obtained from Aby deleting row i and column j .
![Page 94: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/94.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants
Recall:
det
[a bc d
]= ad − bc
det
a b cd e fg h i
= a det
[e fh i
]− b det
[d fg i
]+ c det
[d eg h
]
In general:
det
a1,1 a1,2 · · · a1,na2,1 a2,2 · · · a2,n
...an,1 an,2 · · · an,n
= a1,1D1,1−a1,2D1,2+a1,3D1,3 · · ·±a1,nD1,n
where Di ,j is the determinant of the matrix obtained from Aby deleting row i and column j .
![Page 95: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/95.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants
Recall:
det
[a bc d
]= ad − bc
det
a b cd e fg h i
= a det
[e fh i
]− b det
[d fg i
]+ c det
[d eg h
]
In general:
det
a1,1 a1,2 · · · a1,na2,1 a2,2 · · · a2,n
...an,1 an,2 · · · an,n
= a1,1D1,1−a1,2D1,2+a1,3D1,3 · · ·±a1,nD1,n
where Di ,j is the determinant of the matrix obtained from Aby deleting row i and column j .
![Page 96: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/96.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants
Recall:
det
[a bc d
]= ad − bc
det
a b cd e fg h i
= a det
[e fh i
]− b det
[d fg i
]+ c det
[d eg h
]
In general:
det
a1,1 a1,2 · · · a1,na2,1 a2,2 · · · a2,n
...an,1 an,2 · · · an,n
= a1,1D1,1−a1,2D1,2+a1,3D1,3 · · ·±a1,nD1,n
where Di ,j is the determinant of the matrix obtained from Aby deleting row i and column j .
![Page 97: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/97.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants
Recall:
det
[a bc d
]= ad − bc
det
a b cd e fg h i
= a det
[e fh i
]− b det
[d fg i
]+ c det
[d eg h
]
In general:
det
a1,1 a1,2 · · · a1,na2,1 a2,2 · · · a2,n
...an,1 an,2 · · · an,n
= a1,1D1,1−a1,2D1,2+a1,3D1,3 · · ·±a1,nD1,n
where Di ,j is the determinant of the matrix obtained from Aby deleting row i and column j .
![Page 98: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/98.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Calculate
det
1 2 3 40 1 0 12 0 1 01 0 2 0
= 6
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210
![Page 99: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/99.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Calculate
det
1 2 3 40 1 0 12 0 1 01 0 2 0
= 6
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210
![Page 100: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/100.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Calculate
det
1 2 3 40 1 0 12 0 1 01 0 2 0
= 6
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210
![Page 101: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/101.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Calculate
det
1 2 3 40 1 0 12 0 1 01 0 2 0
= 6
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210
![Page 102: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/102.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Triangular Matrices
Calculate, where ∗ is any number:
det
1 0 0 0 0∗ 2 0 0 0∗ ∗ 3 0 0∗ ∗ ∗ 4 0∗ ∗ ∗ ∗ 5
det
1 ∗ ∗ ∗ ∗0 2 ∗ ∗ ∗0 0 3 ∗ ∗0 0 0 4 ∗0 0 0 0 5
Fact: for any square matrix A, det(A) = det(AT )
![Page 103: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/103.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Triangular Matrices
Calculate, where ∗ is any number:
det
1 0 0 0 0∗ 2 0 0 0∗ ∗ 3 0 0∗ ∗ ∗ 4 0∗ ∗ ∗ ∗ 5
det
1 ∗ ∗ ∗ ∗0 2 ∗ ∗ ∗0 0 3 ∗ ∗0 0 0 4 ∗0 0 0 0 5
Fact: for any square matrix A, det(A) = det(AT )
![Page 104: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/104.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Triangular Matrices
Calculate, where ∗ is any number:
det
1 0 0 0 0∗ 2 0 0 0∗ ∗ 3 0 0∗ ∗ ∗ 4 0∗ ∗ ∗ ∗ 5
det
1 ∗ ∗ ∗ ∗0 2 ∗ ∗ ∗0 0 3 ∗ ∗0 0 0 4 ∗0 0 0 0 5
Fact: for any square matrix A, det(A) = det(AT )
![Page 105: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/105.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Upper Triangular Matrices
Is the determinant of ANY triangular matrix the product of thediagonal entries?
det
[a ∗0 b
]= ab
For 2-by-2 matrices: yes.
The determinant of any triangular matrix (upper or lower) is theproduct of the diagonal entries.
![Page 106: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/106.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Upper Triangular Matrices
Is the determinant of ANY triangular matrix the product of thediagonal entries?
det
[a ∗0 b
]= ab
For 2-by-2 matrices: yes.
The determinant of any triangular matrix (upper or lower) is theproduct of the diagonal entries.
![Page 107: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/107.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Upper Triangular Matrices
Is the determinant of ANY triangular matrix the product of thediagonal entries?
det
[a ∗0 b
]= ab
For 2-by-2 matrices: yes.
The determinant of any triangular matrix (upper or lower) is theproduct of the diagonal entries.
![Page 108: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/108.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Upper Triangular Matrices
Is the determinant of ANY triangular matrix the product of thediagonal entries?
det
[a ∗0 b
]= ab
For 2-by-2 matrices: yes.
det
a ∗ ∗0 b ∗0 0 c
= a det
[b ∗0 c
]− ∗ det
[0 ∗0 c
]+ ∗ det
[0 b0 0
]
The determinant of any triangular matrix (upper or lower) is theproduct of the diagonal entries.
![Page 109: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/109.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Upper Triangular Matrices
Is the determinant of ANY triangular matrix the product of thediagonal entries?
det
[a ∗0 b
]= ab
For 2-by-2 matrices: yes.
det
a ∗ ∗0 b ∗0 0 c
= a det
[b ∗0 c
]︸ ︷︷ ︸triangular
− ∗ det
[0 ∗0 c
]︸ ︷︷ ︸triangular
+ ∗ det
[0 b0 0
]︸ ︷︷ ︸triangular
= a(bc)− ∗(0 · c) + ∗(0 · 0)
= abc
The determinant of any triangular matrix (upper or lower) is theproduct of the diagonal entries.
![Page 110: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/110.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Upper Triangular Matrices
Is the determinant of ANY triangular matrix the product of thediagonal entries?
det
[a ∗0 b
]= ab
For 2-by-2 matrices: yes.
det
a ∗ ∗0 b ∗0 0 c
= a det
[b ∗0 c
]︸ ︷︷ ︸triangular
− ∗ det
[0 ∗0 c
]︸ ︷︷ ︸triangular
+ ∗ det
[0 b0 0
]︸ ︷︷ ︸triangular
= a(bc)− ∗(0 · c) + ∗(0 · 0)
= abc
The determinant of any triangular matrix (upper or lower) is theproduct of the diagonal entries.
![Page 111: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/111.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinants of Upper Triangular Matrices
Is the determinant of ANY triangular matrix the product of thediagonal entries?
det
[a ∗0 b
]= ab
For 2-by-2 matrices: yes.
det
a ∗ ∗0 b ∗0 0 c
= a det
[b ∗0 c
]︸ ︷︷ ︸triangular
− ∗ det
[0 ∗0 c
]︸ ︷︷ ︸triangular
+ ∗ det
[0 b0 0
]︸ ︷︷ ︸triangular
= a(bc)− ∗(0 · c) + ∗(0 · 0)
= abc
The determinant of any triangular matrix (upper or lower) is theproduct of the diagonal entries.
![Page 112: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/112.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
10 9 8 4 120 5 9 7 150 0 1
213
27
0 0 0 2 320 0 0 0 5
= (10)(5)
(1
2
)(2)(5) = 250
Careful: this ONLY works with triangular matrices!
![Page 113: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/113.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
10 9 8 4 120 5 9 7 150 0 1
213
27
0 0 0 2 320 0 0 0 5
= (10)(5)
(1
2
)(2)(5) = 250
Careful: this ONLY works with triangular matrices!
![Page 114: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/114.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
10 9 8 4 120 5 9 7 150 0 1
213
27
0 0 0 2 320 0 0 0 5
= (10)(5)
(1
2
)(2)(5) = 250
Careful: this ONLY works with triangular matrices!
![Page 115: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/115.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Determinant Tricks
Helpful Facts for Calculating the Determinant of a Square MatrixA:
1. If B is obtained from A by multiplying one row of A by theconstant c then detB = c detA.
2. If B is obtained from A by switching two rows of A thendetB = − detA.
3. If B is obtained from A by adding a multiple of one row toanother then detB = detA.
4. det(A) = 0 if and only if A is not invertible
5. For all matrices B of the same size as A,det(AB) = det(A) det(B).
6. det(AT ) = det(A)
Remark: You should understand how the first three lead to the fourth;otherwise, the proofs are optional, found in the notes.
![Page 116: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/116.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Determinant Tricks
Helpful Facts for Calculating the Determinant of a Square MatrixA:
1. If B is obtained from A by multiplying one row of A by theconstant c then detB = c detA.
2. If B is obtained from A by switching two rows of A thendetB = − detA.
3. If B is obtained from A by adding a multiple of one row toanother then detB = detA.
4. det(A) = 0 if and only if A is not invertible
5. For all matrices B of the same size as A,det(AB) = det(A) det(B).
6. det(AT ) = det(A)
Remark: You should understand how the first three lead to the fourth;otherwise, the proofs are optional, found in the notes.
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Determinant Tricks
Helpful Facts for Calculating the Determinant of a Square MatrixA:
1. If B is obtained from A by multiplying one row of A by theconstant c then detB = c detA.
2. If B is obtained from A by switching two rows of A thendetB = − detA.
3. If B is obtained from A by adding a multiple of one row toanother then detB = detA.
4. det(A) = 0 if and only if A is not invertible
5. For all matrices B of the same size as A,det(AB) = det(A) det(B).
6. det(AT ) = det(A)
Remark: You should understand how the first three lead to the fourth;otherwise, the proofs are optional, found in the notes.
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Determinant Tricks
Helpful Facts for Calculating the Determinant of a Square MatrixA:
1. If B is obtained from A by multiplying one row of A by theconstant c then detB = c detA.
2. If B is obtained from A by switching two rows of A thendetB = − detA.
3. If B is obtained from A by adding a multiple of one row toanother then detB = detA.
4. det(A) = 0 if and only if A is not invertible
5. For all matrices B of the same size as A,det(AB) = det(A) det(B).
6. det(AT ) = det(A)
Remark: You should understand how the first three lead to the fourth;otherwise, the proofs are optional, found in the notes.
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
More Determinant Tricks
Helpful Facts for Calculating the Determinant of a Square MatrixA:
1. If B is obtained from A by multiplying one row of A by theconstant c then detB = c detA.
2. If B is obtained from A by switching two rows of A thendetB = − detA.
3. If B is obtained from A by adding a multiple of one row toanother then detB = detA.
4. det(A) = 0 if and only if A is not invertible
5. For all matrices B of the same size as A,det(AB) = det(A) det(B).
6. det(AT ) = det(A)
Remark: You should understand how the first three lead to the fourth;otherwise, the proofs are optional, found in the notes.
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
If A is invertible, then det(A) 6= 0
A is invertible
⇒ we can row-reduce A to the identity matrix⇒ we can row-reduce A to a matrix with determinant 1
• Adding a multiple of one row to another row does notchange the determinant• Swapping two rows multiplies the determinant by −1• Multiplying a row by a constant a multiplies the deter-minant by a
⇒ c det(A) = 1, where c is some constant⇒ det(A) 6= 0
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
If A is invertible, then det(A) 6= 0
A is invertible⇒ we can row-reduce A to the identity matrix
⇒ we can row-reduce A to a matrix with determinant 1• Adding a multiple of one row to another row does notchange the determinant• Swapping two rows multiplies the determinant by −1• Multiplying a row by a constant a multiplies the deter-minant by a
⇒ c det(A) = 1, where c is some constant⇒ det(A) 6= 0
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
If A is invertible, then det(A) 6= 0
A is invertible⇒ we can row-reduce A to the identity matrix⇒ we can row-reduce A to a matrix with determinant 1
• Adding a multiple of one row to another row does notchange the determinant• Swapping two rows multiplies the determinant by −1• Multiplying a row by a constant a multiplies the deter-minant by a
⇒ c det(A) = 1, where c is some constant⇒ det(A) 6= 0
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
If A is invertible, then det(A) 6= 0
A is invertible⇒ we can row-reduce A to the identity matrix⇒ we can row-reduce A to a matrix with determinant 1
• Adding a multiple of one row to another row does notchange the determinant• Swapping two rows multiplies the determinant by −1• Multiplying a row by a constant a multiplies the deter-minant by a
⇒ c det(A) = 1, where c is some constant⇒ det(A) 6= 0
![Page 124: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/124.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If A is invertible, then det(A) 6= 0
A is invertible⇒ we can row-reduce A to the identity matrix⇒ we can row-reduce A to a matrix with determinant 1
• Adding a multiple of one row to another row does notchange the determinant• Swapping two rows multiplies the determinant by −1• Multiplying a row by a constant a multiplies the deter-minant by a
⇒ c det(A) = 1, where c is some constant
⇒ det(A) 6= 0
![Page 125: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/125.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
If A is invertible, then det(A) 6= 0
A is invertible⇒ we can row-reduce A to the identity matrix⇒ we can row-reduce A to a matrix with determinant 1
• Adding a multiple of one row to another row does notchange the determinant• Swapping two rows multiplies the determinant by −1• Multiplying a row by a constant a multiplies the deter-minant by a
⇒ c det(A) = 1, where c is some constant⇒ det(A) 6= 0
![Page 126: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/126.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements areequivalent:
1) Ax = b has exactly one solution for any b .2) Ax = 0 has no nonzero solutions.3) The rank of A is n.4) The reduced form of A has no zeroes along the main diagonal.5) A is invertible6) det(A) 6= 0
statements 1-4invertible
nonzero determinant
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
Is A invertible?
A =
72 9 8 160 4 3 −90 0 5 30 0 0 21
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 1 2 0
10 5 0 110 0 5 30 2 1 1
=?
Calculate: det
1 5 10 150 1 1 10 2 1 20 1 2 1
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
Is A invertible?
A =
72 9 8 160 4 3 −90 0 5 30 0 0 21
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 1 2 0
10 5 0 110 0 5 30 2 1 1
=?
Calculate: det
1 5 10 150 1 1 10 2 1 20 1 2 1
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
Is A invertible?
A =
72 9 8 160 4 3 −90 0 5 30 0 0 21
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 1 2 0
10 5 0 110 0 5 30 2 1 1
=?
Calculate: det
1 5 10 150 1 1 10 2 1 20 1 2 1
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 20 20 01 5 0 22 0 5 10 1 3 1
=?
det
2 0 5 11 5 0 20 10 10 00 1 3 1
=? det
2 0 5 10 10 10 01 5 0 20 1 3 1
=?
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 20 20 01 5 0 22 0 5 10 1 3 1
=?
det
2 0 5 11 5 0 20 10 10 00 1 3 1
=? det
2 0 5 10 10 10 01 5 0 20 1 3 1
=?
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 20 20 01 5 0 22 0 5 10 1 3 1
= −420
det
2 0 5 11 5 0 20 10 10 00 1 3 1
=? det
2 0 5 10 10 10 01 5 0 20 1 3 1
=?
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 20 20 01 5 0 22 0 5 10 1 3 1
= −420
det
2 0 5 11 5 0 20 10 10 00 1 3 1
=? det
2 0 5 10 10 10 01 5 0 20 1 3 1
=?
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 20 20 01 5 0 22 0 5 10 1 3 1
= −420
det
2 0 5 11 5 0 20 10 10 00 1 3 1
= 210 det
2 0 5 10 10 10 01 5 0 20 1 3 1
=?
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 20 20 01 5 0 22 0 5 10 1 3 1
= −420
det
2 0 5 11 5 0 20 10 10 00 1 3 1
= 210 det
2 0 5 10 10 10 01 5 0 20 1 3 1
=?
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
det
0 10 10 01 5 0 22 0 5 10 1 3 1
= −210; det
0 20 20 01 5 0 22 0 5 10 1 3 1
= −420
det
2 0 5 11 5 0 20 10 10 00 1 3 1
= 210 det
2 0 5 10 10 10 01 5 0 20 1 3 1
= −210
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
Suppose detA = 5 for an invertible matrix A. What is det(A−1)?
Suppose A is an n-by-n matrix with determinant 5. What is thedeterminant of 3A?
Suppose A is an n-by-n matrix, and x and y are distinct vectors inRn with Ax = Ay. What is det(A)?
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Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 139: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/139.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 140: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/140.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 141: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/141.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 142: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/142.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−−
det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 143: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/143.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1
R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 144: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/144.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−−
det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 145: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/145.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2
2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 146: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/146.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−−
det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 147: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/147.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2
R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 148: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/148.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 149: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/149.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 150: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/150.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 151: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/151.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 152: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/152.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 153: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/153.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 154: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/154.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 155: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/155.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])
=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 156: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/156.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 157: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/157.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Using Row Reduction to Calculate a Determinant
1 0 41 2 80 1 1
R2−R1−−−−→
1 0 40 2 40 1 1
12R2−−→
1 0 40 1 20 1 1
R3−R2−−−−→
1 0 40 1 20 0 −1
det : −2 R2+R1←−−−− det : −2 2R2←−− det : −1 R3+R2←−−−− det : −1
det
2 2 2 11 1 1 13 5 8 79 6 1 4
= det
0 0 0 −11 1 1 13 5 8 79 6 1 4
= −(−1) det
1 1 13 5 89 6 1
=det
1 1 10 2 50 −3 −8
= det
1 0 01 2 −31 5 −8
= 1det
([2 5−3 −8
])=− 16 + 15 = −1
Is the original 4-by-4 matrix invertible?
![Page 158: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/158.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Suppose a matrix has the following reduced form. Is the matrixinvertible? What is its determinant?1 2 3
0 4 50 0 6
Invertible; determinant unknowable but nonzero
1 2 30 4 50 0 0
Not invertible; determinant 0
1 0 00 1 00 0 1
Invertible; determinant unknowable but nonzero
![Page 159: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/159.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Suppose a matrix has the following reduced form. Is the matrixinvertible? What is its determinant?1 2 3
0 4 50 0 6
Invertible; determinant unknowable but nonzero1 2 3
0 4 50 0 0
Not invertible; determinant 0
1 0 00 1 00 0 1
Invertible; determinant unknowable but nonzero
![Page 160: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/160.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Suppose a matrix has the following reduced form. Is the matrixinvertible? What is its determinant?1 2 3
0 4 50 0 6
Invertible; determinant unknowable but nonzero1 2 3
0 4 50 0 0
Not invertible; determinant 01 0 0
0 1 00 0 1
Invertible; determinant unknowable but nonzero
![Page 161: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/161.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Suppose a matrix has the following reduced form. Is the matrixinvertible? What is its determinant?1 2 3
0 4 50 0 6
Invertible; determinant unknowable but nonzero1 2 3
0 4 50 0 0
Not invertible; determinant 01 0 0
0 1 00 0 1
Invertible; determinant unknowable but nonzero
![Page 162: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/162.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinant Expansion across Alternate Lines
“Line” means “row or column”02+ − + −− + − ++ − + −− + − +
det
9 8 5 6 101 0 0 0 17 0 1 1 18 0 1 1 14 3 5 6 7
det
8 9 5 60 1 1 00 7 1 10 8 1 1
![Page 163: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/163.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinant Expansion across Alternate Lines
“Line” means “row or column”02+ − + −− + − ++ − + −− + − +
det
9 8 5 6 101 0 0 0 17 0 1 1 18 0 1 1 14 3 5 6 7
det
8 9 5 60 1 1 00 7 1 10 8 1 1
![Page 164: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/164.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
Determinant Expansion across Alternate Lines
“Line” means “row or column”02+ − + −− + − ++ − + −− + − +
det
9 8 5 6 101 0 0 0 17 0 1 1 18 0 1 1 14 3 5 6 7
det
8 9 5 60 1 1 00 7 1 10 8 1 1
![Page 165: Course Notes 4.5: Matrix Inverses 4.6: Determinants Outlineelyse/152/2018/8InverseAndDeterminant.pdf · 2018-03-02 · Course Notes 4.5: Matrix Inverses 4.6: Determinants What is](https://reader030.vdocuments.site/reader030/viewer/2022040805/5e4439722ff7880ca605439b/html5/thumbnails/165.jpg)
Course Notes 4.5: Matrix Inverses 4.6: Determinants
More practice
det
2 5 3 40 1 2 04 4 6 9
10 5 7 4