2x2 matrices, determinants and inverses
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2x2 Matrices, Determinants and Inverses. Evaluating Determinants of 2x2 Matrices Using Inverse Matrices to Solve Equations. Evaluating Determinants of 2x2 Matrices. - PowerPoint PPT PresentationTRANSCRIPT
2x2 Matrices, Determinants and Inverses
1. Evaluating Determinants of 2x2 Matrices2. Using Inverse Matrices to Solve Equations
1) Evaluating Determinants of 2x2 Matrices
When you multiply two matrices together, in the order AB or BA, and the result is the identity matrix, then matrices A and B are inverses.
1001
I
Identity matrix for multiplication
1) Evaluating Determinants of 2x2 Matrices
To show two matrices are inverses…AB = I OR BA = I
AA-1 = I OR A-1A = I
Inverse of A Inverse of A
You only have to prove ONE of these.
1) Evaluating Determinants of 2x2 Matrices
Example 1:Show that B is the multiplicative inverse of A.
1713
A
3.07.01.01.0
B
1) Evaluating Determinants of 2x2 Matrices
Example 1:Show that B is the multiplicative inverse of A.
1713
A
3.07.01.01.0
B
3.07.01.01.0
1713
AB
1) Evaluating Determinants of 2x2 Matrices
Example 1:Show that B is the multiplicative inverse of A.
1713
A
3.07.01.01.0
B
3.07.01.01.0
1713
AB
1001
AB
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
1) Evaluating Determinants of 2x2 Matrices
Example 1:Show that B is the multiplicative inverse of A.
1713
A
3.07.01.01.0
B
3.07.01.01.0
1713
AB
1713
3.07.01.01.0
BA
1001
AB
Check by multiplying BA…answer should be the same
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
1) Evaluating Determinants of 2x2 Matrices
Example 1:Show that B is the multiplicative inverse of A.
1713
A
3.07.01.01.0
B
3.07.01.01.0
1713
AB
1713
3.07.01.01.0
BA
1001
AB
1001
BA
Check by multiplying BA…answer should be the same
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
1) Evaluating Determinants of 2x2 Matrices
Example 2:Show that the matrices are multiplicative inverses.
8352
A
2358
B
1) Evaluating Determinants of 2x2 Matrices
Example 2:Show that the matrices are multiplicative inverses.
8352
A
2358
B
8352
2358
BA
1001
BA
BA = I. Therefore, B is the inverse of A and A is the inverse of B.
The determinant is used to tell us if an inverse exists.
If det ≠ 0, an inverse exists.
If det = 0, no inverse exists. A Matrix with a determinant of zero is called a SINGULAR matrix
1) Evaluating Determinants of 2x2 Matrices
1) Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
dcba
A dcba
A det
1) Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
dcba
A dcba
A det
dcba
Multiply along the diagonal
1) Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
dcba
A dcba
A det
dcba
bcad
Take the product of the leading diagonal, and subtract the product of the non-leading diagonal
Equation to find the determinant
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
9587
det
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
9587
det
9587
det
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
9587
det
9587
9587
det
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
9587
det
9587
)5)(8()9)(7(
23
det = -23Therefore, there is an inverse.
9587
det
1) Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.
2424
det
1) Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.
2424
det
)2)(4()2)(4( 0
2424
det
1) Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.
2424
det
)2)(4()2)(4( 0
2424
det
det = 0
Therefore, there is no inverse.
1) Evaluating Determinants of 2x2 Matrices
How do you know if a matrix has an inverse AND what that inverse is?Given , the inverse of A is given by:
acbd
AA
det11
Equation to find an inverse matrix
This is called the adjoint matrix. It is formed by interchanging elements in the leading diagonal and negating elements in the non-leading diagonal
dcba
A
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
M
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MStep 1: Find det M
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MStep 1: Find det M
)5)(2()4)(2( bcad
2
det M = -2, the inverse of M exists.
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MStep 2: Find the adjoint matrix. i.e
acbd
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MChange signs
Step 2: Find the adjoint matrix. i.e
acbd
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MChange signs
?52?
Step 2: Find the adjoint matrix. i.e
acbd
Adjoint of M
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MChange positions
?52?
Step 2: Find the adjoint matrix. i.e
acbd
Adjoint of M
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MStep 2: Find the adjoint matrix. i.e
acbd
2524
Change positions
Adjoint of M
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MStep 3: Use the equation to find the inverse.
2524
211M
ofMAdjoM
M intdet
11
1) Evaluating Determinants of 2x2 Matrices
Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.
4522
MStep 3: Use the equation to find the inverse.
2524
211M
15.2121M
1) Evaluating Determinants of 2x2 Matrices
Example 2:Determine whether the matrix has an inverse. If an inverse exists, find it.
3142
1) Evaluating Determinants of 2x2 Matrices
Example 2:Determine whether the matrix has an inverse. If an inverse exists, find it.
3142
)1)(4()3)(2( bcad
2
3142
3142
det
1) Evaluating Determinants of 2x2 Matrices
Example 2:Determine whether the matrix has an inverse. If an inverse exists, find it.
3142
2143
211A
15.025.11A