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3.9 Determinants - PowerPoint PPT Presentation

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. a. b. a. b. det = = ad - bc. c. d. c. d.

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• Given a square matrix Aits 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

a

b

a

b

det = = ad - bc

c

d

c

d

• For larger matrices the definition is more complicated

2

det = = (-5)(0) – (2)(-2) = 4

-2

0

Determinants 2x2 examples

-5

1

1

1

1

2

2

2

2

2

det = = (1)(4) – (2)(3) = -2

-2

3

2

2

3

4

0

4

4

4

det = = (1)(4) – (2)(2) = 0

• 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

M11 : remove row 1, col 1

1

1

-2

A =

-1

2

3

2

3

M11 =

2

7

0

7

0

• 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

M12 : remove row 1, col 2

1

1

-2

A =

-1

2

3

-1

3

M12 =

2

7

0

2

0

• 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

M13 : remove row 1, col 3

1

1

-2

A =

-1

2

3

-1

2

M13 =

2

7

0

2

7

• 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

M21 : remove row 2, col 1

-2

1

1

A =

-1

2

3

1

-2

M21 =

2

7

0

7

0

• 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

M22 : remove row 2, col 2

-2

1

1

A =

-1

2

3

1

-2

M22 =

2

7

0

2

0

• 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

M23 : remove row 2, col 3

-2

1

1

A =

-1

2

3

1

1

M23 =

2

7

0

2

7

• 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

M31 : remove row 3, col 1

-2

1

1

A =

-1

2

3

1

-2

M31 =

2

7

0

2

3

• 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

M32 : remove row 3, col 2

-2

1

1

A =

-1

2

3

1

-2

M32 =

2

7

0

-1

3

• 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

M33 : remove row 3, col 3

-2

1

1

A =

-1

2

3

1

1

M33 =

2

7

0

-1

2

a13

a11

a12

• For a matrix

a21

a23

A =

a22

a32

a31

a33

• Its determinant is given by

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

• From the formula for a 2x2 matrix:

a22

a23

|M11|= = a22a33 - a23a32

a32

a33

a13

a11

a12

• For a matrix

a21

a23

A =

a22

a32

a31

a33

• Its determinant is given by

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

• From the formula for a 2x2 matrix:

a21

a23

|M12|= = a21a33 - a23a31

a31

a33

a13

a11

a12

• For a matrix

a21

a23

A =

a22

a32

a31

a33

• Its determinant is given by

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

• From the formula for a 2x2 matrix:

a21

a22

|M13|= = a21a32 - a31a22

a31

a32

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

1

1

-2

-1

3

2

-1

3

2

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

A =

-1

2

3

2

0

7

2

0

7

2

7

0

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

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

0

1

3

5

1

3

5

1

3

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

B =

5

3

1

-1

0

2

-1

0

2

-1

2

0

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

a13

a11

a12

• For the matrix

a21

a23

A =

a22

a32

a31

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

a13

a11

a12

• For the matrix

a21

a23

A =

a22

a32

a31

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|

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

+

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

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

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

-

+

-

-

+

+

-

+

a13

+

a11

a12

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

a21

a23

A =

a22

a32

a31

a33

• E.g. using the first column:

-

+

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

-

-

+

+

-

+

• Using the first row:

1

1

-2

0

3

2

0

3

2

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

A =

0

2

3

0

1

1

0

1

1

0

1

1

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

• However, using the first column:

1

1

-2

2

3

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

A =

0

2

3

1

1

0

1

1

3.9.2 A general formula for determinants determinants. e.g.

• 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 determinants. e.g.

• 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|= aijCijfor any j=1,2,...,n

• Or,

|A|= aijCijfor any i=1,2,...,n