3.9 Determinants

• Given a square matrix A its determinant is a real number associated with the matrix.

• The determinant of A is written:

det(A) or |A|• For a 2x2 matrix, the definition is

det = = ad - bcac

bd

• For larger matrices the definition is more complicated

ac

bd

13

24

det = = (1)(4) – (2)(3) = -213

24

-5-2

20

det = = (-5)(0) – (2)(-2) = 4-5-2

20

12

24

det = = (1)(4) – (2)(2) = 012

24

Determinants 2x2 examples

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

Determinants

-21 1-1 22 7

3 0

A =M11 =

27

30

M11 : remove row 1, col 1

-21 1-1 22 7

3 0

A =M12 =

-12

30

M12 : remove row 1, col 2

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

-21 1-1 22 7

3 0

A =M13 =

-12

27

M13 : remove row 1, col 3

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

-21 1-1 22 7

3 0

A =M21 =

17

-20

M21 : remove row 2, col 1

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

-21 1-1 22 7

3 0

A =M22 =

12

-20

M22 : remove row 2, col 2

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

-21 1-1 22 7

3 0

A =M23 =

12

17

M23 : remove row 2, col 3

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

-21 1-1 22 7

3 0

A =M31 =

12

-23

M31 : remove row 3, col 1

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

-21 1-1 22 7

3 0

A =M32 =

1-1

-23

M32 : remove row 3, col 2

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

-21 1-1 22 7

3 0

A =M33 =

1-1

12

M33 : remove row 3, col 3

Determinants

• To define det(A) for larger matrices, we will need the definition of a minor Mij

• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A

• For a matrix a11

A = a21

a12

a31

a22

a32

a13

a23

a33

• Its determinant is given by

|A| = a11|M11| - a12|M12| + a13|M13|

• From the formula for a 2x2 matrix:

|M11|= = a22a33 - a23a32 a22

a32

a23

a33

3.9.1 The formula for a 3x3 matrix

• For a matrix a11

A = a21

a12

a31

a22

a32

a13

a23

a33

• Its determinant is given by

|A| = a11|M11| - a12|M12| + a13|M13|

• From the formula for a 2x2 matrix:

|M12|= = a21a33 - a23a31 a21

a31

a23

a33

3.9.1 The formula for a 3x3 matrix

• For a matrix a11

A = a21

a12

a31

a22

a32

a13

a23

a33

• Its determinant is given by

|A| = a11|M11| - a12|M12| + a13|M13|

• From the formula for a 2x2 matrix:

|M13|= = a21a32 - a31a22 a21

a31

a22

a32

3.9.1 The formula for a 3x3 matrix

-21 1-1 22 7

3 0

A =

= 1x(-21) -1x(-6) +(-2)x(-11) = 7

|A|= 1x - 1x + (-2)27

30

-12

30

-12

27

|A| = 1x|M11| - 1x|M12| + (-2)x|M13|

3x3 Example

30 15 3-1 2

1 0

B =

= 0x(-2) -1x(1) +(3)x(13) = 38

|B|= 0x - 1x + 3 x32

10

5-1

10

5-1

32

|B| = 0x|M11| - 1x|M12| + 3x|M13|

3x3 Example

• For the matrix a11

A = a21

a12

a31

a22

a32

a13

a23

a33

• We used the top row to calculate the determinant:

|A| = a11|M11| - a12|M12| + a13|M13| • However, we could equally have used any row of

the matrix and performed a similar calculation

3.9.1 The formula for a 3x3 matrix

• For the matrix a11

A = a21

a12

a31

a22

a32

a13

a23

a33

• Using the top row:|A| = a11|M11| - a12|M12| + a13|M13|

• Using the second row |A| = -a21|M21| + a22|M22| - a23|M23|

• Using the third row

|A| = a31|M31| - a32|M32| + a33|M33|

3.9.1 The formula for a 3x3 matrix

|A| = a11|M11| - a12|M12| + a13|M13|

• Notice the changing signs depending on what row we use:

= -a21|M21| + a22|M22| - a23|M23|

= a31|M31| - a32|M32| + a33|M33|

+ +

++

+-

--

-

3.9.1 The formula for a 3x3 matrix

• Equally, we could have used any column as long as we follow the signs pattern

+ +

++

+-

--

-

• E.g. using the first column:

|A| = a11|M11| - a21|M21| + a31|M31|

a11

A = a21

a12

a31

a22

a32

a13

a23

a33

3.9.1 The formula for a 3x3 matrix

• This choice sometimes makes it a bit easier to calculate determinants. e.g.

-21 1 0 20 1

3 1

A =

|A|= 1x - 1x + (-2) x21

31

0 0

31

0 0

21

= 1x(-1) -1x(0) + (-2)x(0) = -1

• Using the first row:

• This choice sometimes makes it a bit easier to calculate determinants. e.g.

-21 1 0 20 1

3 1

A =

|A|= 1x - 0 + 0 = 1x(-1) = -1 21

31

• However, using the first column:

• For a 4x4 matrix we add up minors like the 3x3 case, and again use the same signs pattern

+ +

++

+-

--

-+ +

+

- --

-

• Notice that if we think of the signs pattern as a matrix, then it can be written as (-1)i+j

3.9.2 A general formula for determinants

• For a nxn matrix A=(aij) the co-factors of A are defined by

Cij:= (-1)i+j|Mij|• The determinant of A is given by the formula

|A|= aij Cij for any j=1,2,...,n

n

i 1• Or,

|A|= aij Cij for any i=1,2,...,n

n

j 1

3.9.2 A general formula for determinants