Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 6 & 4 & 2 5 & 1 & 8 6 & 8 & 7 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 6 & 4 & 2 \\ 5 & 1 & 8 \\ 6 & 8 & 7 \end{array}]$

Step 1

Consider the corresponding sign chart.

⎡ ⎢ ⎣ \begin{matrix} + & - & + - & + & - + & - & + \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array}]$

Step 2

Use the sign chart and the given matrix to find the cofactor of each element.

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Step 2.1

Calculate the minor for element

a_{11}

$a_{11}$ .

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Step 2.1.1

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 1 & 8 8 & 7 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 1 & 8 \\ 8 & 7 \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{11} = 1 \cdot 7 - 8 \cdot 8

$a_{11} = 1 \cdot 7 - 8 \cdot 8$

Step 2.1.2.2

Simplify the determinant.

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Step 2.1.2.2.1

Simplify each term.

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Step 2.1.2.2.1.1

Multiply

7

$7$ by

1

$1$ .

a_{11} = 7 - 8 \cdot 8

$a_{11} = 7 - 8 \cdot 8$

Step 2.1.2.2.1.2

Multiply

- 8

$- 8$ by

8

$8$ .

a_{11} = 7 - 64

$a_{11} = 7 - 64$

a_{11} = 7 - 64

$a_{11} = 7 - 64$

Step 2.1.2.2.2

Subtract

64

$64$ from

7

$7$ .

a_{11} = - 57

$a_{11} = - 57$

a_{11} = - 57

$a_{11} = - 57$

a_{11} = - 57

$a_{11} = - 57$

a_{11} = - 57

$a_{11} = - 57$

Step 2.2

Calculate the minor for element

a_{12}

$a_{12}$ .

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Step 2.2.1

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 5 & 8 6 & 7 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 5 & 8 \\ 6 & 7 \end{array} |$

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{12} = 5 \cdot 7 - 6 \cdot 8

$a_{12} = 5 \cdot 7 - 6 \cdot 8$

Step 2.2.2.2

Simplify the determinant.

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Step 2.2.2.2.1

Simplify each term.

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Step 2.2.2.2.1.1

Multiply

5

$5$ by

7

$7$ .

a_{12} = 35 - 6 \cdot 8

$a_{12} = 35 - 6 \cdot 8$

Step 2.2.2.2.1.2

Multiply

- 6

$- 6$ by

8

$8$ .

a_{12} = 35 - 48

$a_{12} = 35 - 48$

a_{12} = 35 - 48

$a_{12} = 35 - 48$

Step 2.2.2.2.2

Subtract

48

$48$ from

35

$35$ .

a_{12} = - 13

$a_{12} = - 13$

a_{12} = - 13

$a_{12} = - 13$

a_{12} = - 13

$a_{12} = - 13$

a_{12} = - 13

$a_{12} = - 13$

Step 2.3

Calculate the minor for element

a_{13}

$a_{13}$ .

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Step 2.3.1

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

∣ ∣ ∣ \begin{matrix} 5 & 1 6 & 8 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 5 & 1 \\ 6 & 8 \end{array} |$

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{13} = 5 \cdot 8 - 6 \cdot 1

$a_{13} = 5 \cdot 8 - 6 \cdot 1$

Step 2.3.2.2

Simplify the determinant.

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Step 2.3.2.2.1

Simplify each term.

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Step 2.3.2.2.1.1

Multiply

5

$5$ by

8

$8$ .

a_{13} = 40 - 6 \cdot 1

$a_{13} = 40 - 6 \cdot 1$

Step 2.3.2.2.1.2

Multiply

- 6

$- 6$ by

1

$1$ .

a_{13} = 40 - 6

$a_{13} = 40 - 6$

a_{13} = 40 - 6

$a_{13} = 40 - 6$

Step 2.3.2.2.2

Subtract

6

$6$ from

40

$40$ .

a_{13} = 34

$a_{13} = 34$

a_{13} = 34

$a_{13} = 34$

a_{13} = 34

$a_{13} = 34$

a_{13} = 34

$a_{13} = 34$

Step 2.4

Calculate the minor for element

a_{21}

$a_{21}$ .

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Step 2.4.1

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 4 & 2 8 & 7 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 4 & 2 \\ 8 & 7 \end{array} |$

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{21} = 4 \cdot 7 - 8 \cdot 2

$a_{21} = 4 \cdot 7 - 8 \cdot 2$

Step 2.4.2.2

Simplify the determinant.

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Step 2.4.2.2.1

Simplify each term.

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Step 2.4.2.2.1.1

Multiply

4

$4$ by

7

$7$ .

a_{21} = 28 - 8 \cdot 2

$a_{21} = 28 - 8 \cdot 2$

Step 2.4.2.2.1.2

Multiply

- 8

$- 8$ by

2

$2$ .

a_{21} = 28 - 16

$a_{21} = 28 - 16$

a_{21} = 28 - 16

$a_{21} = 28 - 16$

Step 2.4.2.2.2

Subtract

16

$16$ from

28

$28$ .

a_{21} = 12

$a_{21} = 12$

a_{21} = 12

$a_{21} = 12$

a_{21} = 12

$a_{21} = 12$

a_{21} = 12

$a_{21} = 12$

Step 2.5

Calculate the minor for element

a_{22}

$a_{22}$ .

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Step 2.5.1

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 6 & 2 6 & 7 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 6 & 2 \\ 6 & 7 \end{array} |$

Step 2.5.2

Evaluate the determinant.

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Step 2.5.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{22} = 6 \cdot 7 - 6 \cdot 2

$a_{22} = 6 \cdot 7 - 6 \cdot 2$

Step 2.5.2.2

Simplify the determinant.

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Step 2.5.2.2.1

Simplify each term.

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Step 2.5.2.2.1.1

Multiply

6

$6$ by

7

$7$ .

a_{22} = 42 - 6 \cdot 2

$a_{22} = 42 - 6 \cdot 2$

Step 2.5.2.2.1.2

Multiply

- 6

$- 6$ by

2

$2$ .

a_{22} = 42 - 12

$a_{22} = 42 - 12$

a_{22} = 42 - 12

$a_{22} = 42 - 12$

Step 2.5.2.2.2

Subtract

12

$12$ from

42

$42$ .

a_{22} = 30

$a_{22} = 30$

a_{22} = 30

$a_{22} = 30$

a_{22} = 30

$a_{22} = 30$

a_{22} = 30

$a_{22} = 30$

Step 2.6

Calculate the minor for element

a_{23}

$a_{23}$ .

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Step 2.6.1

The minor for

a_{23}

$a_{23}$ is the determinant with row

2

$2$ and column

3

$3$ deleted.

∣ ∣ ∣ \begin{matrix} 6 & 4 6 & 8 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 6 & 4 \\ 6 & 8 \end{array} |$

Step 2.6.2

Evaluate the determinant.

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Step 2.6.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{23} = 6 \cdot 8 - 6 \cdot 4

$a_{23} = 6 \cdot 8 - 6 \cdot 4$

Step 2.6.2.2

Simplify the determinant.

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Step 2.6.2.2.1

Simplify each term.

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Step 2.6.2.2.1.1

Multiply

6

$6$ by

8

$8$ .

a_{23} = 48 - 6 \cdot 4

$a_{23} = 48 - 6 \cdot 4$

Step 2.6.2.2.1.2

Multiply

- 6

$- 6$ by

4

$4$ .

a_{23} = 48 - 24

$a_{23} = 48 - 24$

a_{23} = 48 - 24

$a_{23} = 48 - 24$

Step 2.6.2.2.2

Subtract

24

$24$ from

48

$48$ .

a_{23} = 24

$a_{23} = 24$

a_{23} = 24

$a_{23} = 24$

a_{23} = 24

$a_{23} = 24$

a_{23} = 24

$a_{23} = 24$

Step 2.7

Calculate the minor for element

a_{31}

$a_{31}$ .

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Step 2.7.1

The minor for

a_{31}

$a_{31}$ is the determinant with row

3

$3$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 4 & 2 1 & 8 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 4 & 2 \\ 1 & 8 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{31} = 4 \cdot 8 - 1 \cdot 2

$a_{31} = 4 \cdot 8 - 1 \cdot 2$

Step 2.7.2.2

Simplify the determinant.

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Step 2.7.2.2.1

Simplify each term.

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Step 2.7.2.2.1.1

Multiply

4

$4$ by

8

$8$ .

a_{31} = 32 - 1 \cdot 2

$a_{31} = 32 - 1 \cdot 2$

Step 2.7.2.2.1.2

Multiply

- 1

$- 1$ by

2

$2$ .

a_{31} = 32 - 2

$a_{31} = 32 - 2$

a_{31} = 32 - 2

$a_{31} = 32 - 2$

Step 2.7.2.2.2

Subtract

2

$2$ from

32

$32$ .

a_{31} = 30

$a_{31} = 30$

a_{31} = 30

$a_{31} = 30$

a_{31} = 30

$a_{31} = 30$

a_{31} = 30

$a_{31} = 30$

Step 2.8

Calculate the minor for element

a_{32}

$a_{32}$ .

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Step 2.8.1

The minor for

a_{32}

$a_{32}$ is the determinant with row

3

$3$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 6 & 2 5 & 8 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 6 & 2 \\ 5 & 8 \end{array} |$

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{32} = 6 \cdot 8 - 5 \cdot 2

$a_{32} = 6 \cdot 8 - 5 \cdot 2$

Step 2.8.2.2

Simplify the determinant.

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Step 2.8.2.2.1

Simplify each term.

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Step 2.8.2.2.1.1

Multiply

6

$6$ by

8

$8$ .

a_{32} = 48 - 5 \cdot 2

$a_{32} = 48 - 5 \cdot 2$

Step 2.8.2.2.1.2

Multiply

- 5

$- 5$ by

2

$2$ .

a_{32} = 48 - 10

$a_{32} = 48 - 10$

a_{32} = 48 - 10

$a_{32} = 48 - 10$

Step 2.8.2.2.2

Subtract

10

$10$ from

48

$48$ .

a_{32} = 38

$a_{32} = 38$

a_{32} = 38

$a_{32} = 38$

a_{32} = 38

$a_{32} = 38$

a_{32} = 38

$a_{32} = 38$

Step 2.9

Calculate the minor for element

a_{33}

$a_{33}$ .

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Step 2.9.1

The minor for

a_{33}

$a_{33}$ is the determinant with row

3

$3$ and column

3

$3$ deleted.

∣ ∣ ∣ \begin{matrix} 6 & 4 5 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 6 & 4 \\ 5 & 1 \end{array} |$

Step 2.9.2

Evaluate the determinant.

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Step 2.9.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{33} = 6 \cdot 1 - 5 \cdot 4

$a_{33} = 6 \cdot 1 - 5 \cdot 4$

Step 2.9.2.2

Simplify the determinant.

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Step 2.9.2.2.1

Simplify each term.

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Step 2.9.2.2.1.1

Multiply

6

$6$ by

1

$1$ .

a_{33} = 6 - 5 \cdot 4

$a_{33} = 6 - 5 \cdot 4$

Step 2.9.2.2.1.2

Multiply

- 5

$- 5$ by

4

$4$ .

a_{33} = 6 - 20

$a_{33} = 6 - 20$

a_{33} = 6 - 20

$a_{33} = 6 - 20$

Step 2.9.2.2.2

Subtract

20

$20$ from

6

$6$ .

a_{33} = - 14

$a_{33} = - 14$

Step 2.10

The cofactor matrix is a matrix of the minors with the sign changed for the elements in the

$-$ positions on the sign chart.

$[\begin{array}{c} - 57 & 13 & 34 \\ - 12 & 30 & - 24 \\ 30 & - 38 & - 14 \end{array}]$