Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 4 & 0 & - 2 2 & 3 & 4 5 & - 7 & 6 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 4 & 0 & - 2 \\ 2 & 3 & 4 \\ 5 & - 7 & 6 \end{array}]$

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} 3 & 4 - 7 & 6 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 3 & 4 \\ - 7 & 6 \end{array} |$

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} = 3 \cdot 6 - (- 7 \cdot 4)

$a_{11} = 3 \cdot 6 - (- 7 \cdot 4)$

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

3

$3$ by

6

$6$ .

a_{11} = 18 - (- 7 \cdot 4)

$a_{11} = 18 - (- 7 \cdot 4)$

Step 2.1.2.2.1.2

Multiply

- (- 7 \cdot 4)

$- (- 7 \cdot 4)$ .

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Step 2.1.2.2.1.2.1

Multiply

- 7

$- 7$ by

4

$4$ .

a_{11} = 18 - - 28

$a_{11} = 18 - - 28$

Step 2.1.2.2.1.2.2

Multiply

- 1

$- 1$ by

- 28

$- 28$ .

a_{11} = 18 + 28

$a_{11} = 18 + 28$

a_{11} = 18 + 28

$a_{11} = 18 + 28$

a_{11} = 18 + 28

$a_{11} = 18 + 28$

Step 2.1.2.2.2

Add

18

$18$ and

28

$28$ .

a_{11} = 46

$a_{11} = 46$

a_{11} = 46

$a_{11} = 46$

a_{11} = 46

$a_{11} = 46$

a_{11} = 46

$a_{11} = 46$

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} 2 & 4 5 & 6 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 4 \\ 5 & 6 \end{array} |$

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} = 2 \cdot 6 - 5 \cdot 4

$a_{12} = 2 \cdot 6 - 5 \cdot 4$

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

2

$2$ by

6

$6$ .

a_{12} = 12 - 5 \cdot 4

$a_{12} = 12 - 5 \cdot 4$

Step 2.2.2.2.1.2

Multiply

- 5

$- 5$ by

4

$4$ .

a_{12} = 12 - 20

$a_{12} = 12 - 20$

a_{12} = 12 - 20

$a_{12} = 12 - 20$

Step 2.2.2.2.2

Subtract

20

$20$ from

12

$12$ .

a_{12} = - 8

$a_{12} = - 8$

a_{12} = - 8

$a_{12} = - 8$

a_{12} = - 8

$a_{12} = - 8$

a_{12} = - 8

$a_{12} = - 8$

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} 2 & 3 5 & - 7 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 3 \\ 5 & - 7 \end{array} |$

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} = 2 \cdot - 7 - 5 \cdot 3

$a_{13} = 2 \cdot - 7 - 5 \cdot 3$

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

2

$2$ by

- 7

$- 7$ .

a_{13} = - 14 - 5 \cdot 3

$a_{13} = - 14 - 5 \cdot 3$

Step 2.3.2.2.1.2

Multiply

- 5

$- 5$ by

3

$3$ .

a_{13} = - 14 - 15

$a_{13} = - 14 - 15$

a_{13} = - 14 - 15

$a_{13} = - 14 - 15$

Step 2.3.2.2.2

Subtract

15

$15$ from

- 14

$- 14$ .

a_{13} = - 29

$a_{13} = - 29$

a_{13} = - 29

$a_{13} = - 29$

a_{13} = - 29

$a_{13} = - 29$

a_{13} = - 29

$a_{13} = - 29$

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} 0 & - 2 - 7 & 6 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 0 & - 2 \\ - 7 & 6 \end{array} |$

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} = 0 \cdot 6 - (- 7 \cdot - 2)

$a_{21} = 0 \cdot 6 - (- 7 \cdot - 2)$

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

0

$0$ by

6

$6$ .

a_{21} = 0 - (- 7 \cdot - 2)

$a_{21} = 0 - (- 7 \cdot - 2)$

Step 2.4.2.2.1.2

Multiply

- (- 7 \cdot - 2)

$- (- 7 \cdot - 2)$ .

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Step 2.4.2.2.1.2.1

Multiply

- 7

$- 7$ by

- 2

$- 2$ .

a_{21} = 0 - 1 \cdot 14

$a_{21} = 0 - 1 \cdot 14$

Step 2.4.2.2.1.2.2

Multiply

- 1

$- 1$ by

14

$14$ .

a_{21} = 0 - 14

$a_{21} = 0 - 14$

a_{21} = 0 - 14

$a_{21} = 0 - 14$

a_{21} = 0 - 14

$a_{21} = 0 - 14$

Step 2.4.2.2.2

Subtract

14

$14$ from

0

$0$ .

a_{21} = - 14

$a_{21} = - 14$

a_{21} = - 14

$a_{21} = - 14$

a_{21} = - 14

$a_{21} = - 14$

a_{21} = - 14

$a_{21} = - 14$

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} 4 & - 2 5 & 6 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 4 & - 2 \\ 5 & 6 \end{array} |$

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} = 4 \cdot 6 - 5 \cdot - 2

$a_{22} = 4 \cdot 6 - 5 \cdot - 2$

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

4

$4$ by

6

$6$ .

a_{22} = 24 - 5 \cdot - 2

$a_{22} = 24 - 5 \cdot - 2$

Step 2.5.2.2.1.2

Multiply

- 5

$- 5$ by

- 2

$- 2$ .

a_{22} = 24 + 10

$a_{22} = 24 + 10$

a_{22} = 24 + 10

$a_{22} = 24 + 10$

Step 2.5.2.2.2

Add

24

$24$ and

10

$10$ .

a_{22} = 34

$a_{22} = 34$

a_{22} = 34

$a_{22} = 34$

a_{22} = 34

$a_{22} = 34$

a_{22} = 34

$a_{22} = 34$

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} 4 & 0 5 & - 7 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 4 & 0 \\ 5 & - 7 \end{array} |$

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} = 4 \cdot - 7 - 5 \cdot 0

$a_{23} = 4 \cdot - 7 - 5 \cdot 0$

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

4

$4$ by

- 7

$- 7$ .

a_{23} = - 28 - 5 \cdot 0

$a_{23} = - 28 - 5 \cdot 0$

Step 2.6.2.2.1.2

Multiply

- 5

$- 5$ by

0

$0$ .

a_{23} = - 28 + 0

$a_{23} = - 28 + 0$

a_{23} = - 28 + 0

$a_{23} = - 28 + 0$

Step 2.6.2.2.2

Add

- 28

$- 28$ and

0

$0$ .

a_{23} = - 28

$a_{23} = - 28$

a_{23} = - 28

$a_{23} = - 28$

a_{23} = - 28

$a_{23} = - 28$

a_{23} = - 28

$a_{23} = - 28$

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} 0 & - 2 3 & 4 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 0 & - 2 \\ 3 & 4 \end{array} |$

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} = 0 \cdot 4 - 3 \cdot - 2

$a_{31} = 0 \cdot 4 - 3 \cdot - 2$

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

0

$0$ by

4

$4$ .

a_{31} = 0 - 3 \cdot - 2

$a_{31} = 0 - 3 \cdot - 2$

Step 2.7.2.2.1.2

Multiply

- 3

$- 3$ by

- 2

$- 2$ .

a_{31} = 0 + 6

$a_{31} = 0 + 6$

a_{31} = 0 + 6

$a_{31} = 0 + 6$

Step 2.7.2.2.2

Add

0

$0$ and

6

$6$ .

a_{31} = 6

$a_{31} = 6$

a_{31} = 6

$a_{31} = 6$

a_{31} = 6

$a_{31} = 6$

a_{31} = 6

$a_{31} = 6$

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} 4 & - 2 2 & 4 \end{matrix} ∣ ∣ ∣

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

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} = 4 \cdot 4 - 2 \cdot - 2

$a_{32} = 4 \cdot 4 - 2 \cdot - 2$

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

4

$4$ by

4

$4$ .

a_{32} = 16 - 2 \cdot - 2

$a_{32} = 16 - 2 \cdot - 2$

Step 2.8.2.2.1.2

Multiply

- 2

$- 2$ by

- 2

$- 2$ .

a_{32} = 16 + 4

$a_{32} = 16 + 4$

a_{32} = 16 + 4

$a_{32} = 16 + 4$

Step 2.8.2.2.2

Add

16

$16$ and

4

$4$ .

a_{32} = 20

$a_{32} = 20$

a_{32} = 20

$a_{32} = 20$

a_{32} = 20

$a_{32} = 20$

a_{32} = 20

$a_{32} = 20$

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} 4 & 0 2 & 3 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 4 & 0 \\ 2 & 3 \end{array} |$

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} = 4 \cdot 3 - 2 \cdot 0

$a_{33} = 4 \cdot 3 - 2 \cdot 0$

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

4

$4$ by

3

$3$ .

a_{33} = 12 - 2 \cdot 0

$a_{33} = 12 - 2 \cdot 0$

Step 2.9.2.2.1.2

Multiply

- 2

$- 2$ by

0

$0$ .

a_{33} = 12 + 0

$a_{33} = 12 + 0$

a_{33} = 12 + 0

$a_{33} = 12 + 0$

Step 2.9.2.2.2

Add

12

$12$ and

0

$0$ .

a_{33} = 12

$a_{33} = 12$

a_{33} = 12

$a_{33} = 12$

a_{33} = 12

$a_{33} = 12$

a_{33} = 12

$a_{33} = 12$

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{matrix} 46 & 8 & - 29 14 & 34 & 28 6 & - 20 & 12 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 46 & 8 & - 29 \\ 14 & 34 & 28 \\ 6 & - 20 & 12 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 46 & 8 & - 29 14 & 34 & 28 6 & - 20 & 12 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 46 & 8 & - 29 \\ 14 & 34 & 28 \\ 6 & - 20 & 12 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

⎡ ⎢ ⎣ \begin{matrix} 46 & 14 & 6 8 & 34 & - 20 - 29 & 28 & 12 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 46 & 14 & 6 \\ 8 & 34 & - 20 \\ - 29 & 28 & 12 \end{array}]$