Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} - 63 & 30 & 24 33 & - 15 & - 9 - 45 & 30 & 15 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} - 63 & 30 & 24 \\ 33 & - 15 & - 9 \\ - 45 & 30 & 15 \end{array}]$

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} - 15 & - 9 30 & 15 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} - 15 & - 9 \\ 30 & 15 \end{array} |$

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} = - 15 \cdot 15 - 30 \cdot - 9

$a_{11} = - 15 \cdot 15 - 30 \cdot - 9$

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

- 15

$- 15$ by

15

$15$ .

a_{11} = - 225 - 30 \cdot - 9

$a_{11} = - 225 - 30 \cdot - 9$

Step 2.1.2.2.1.2

Multiply

- 30

$- 30$ by

- 9

$- 9$ .

a_{11} = - 225 + 270

$a_{11} = - 225 + 270$

a_{11} = - 225 + 270

$a_{11} = - 225 + 270$

Step 2.1.2.2.2

Add

- 225

$- 225$ and

270

$270$ .

a_{11} = 45

$a_{11} = 45$

a_{11} = 45

$a_{11} = 45$

a_{11} = 45

$a_{11} = 45$

a_{11} = 45

$a_{11} = 45$

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} 33 & - 9 - 45 & 15 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 33 & - 9 \\ - 45 & 15 \end{array} |$

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} = 33 \cdot 15 - (- 45 \cdot - 9)

$a_{12} = 33 \cdot 15 - (- 45 \cdot - 9)$

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

33

$33$ by

15

$15$ .

a_{12} = 495 - (- 45 \cdot - 9)

$a_{12} = 495 - (- 45 \cdot - 9)$

Step 2.2.2.2.1.2

Multiply

- (- 45 \cdot - 9)

$- (- 45 \cdot - 9)$ .

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Step 2.2.2.2.1.2.1

Multiply

- 45

$- 45$ by

- 9

$- 9$ .

a_{12} = 495 - 1 \cdot 405

$a_{12} = 495 - 1 \cdot 405$

Step 2.2.2.2.1.2.2

Multiply

- 1

$- 1$ by

405

$405$ .

a_{12} = 495 - 405

$a_{12} = 495 - 405$

a_{12} = 495 - 405

$a_{12} = 495 - 405$

a_{12} = 495 - 405

$a_{12} = 495 - 405$

Step 2.2.2.2.2

Subtract

405

$405$ from

495

$495$ .

a_{12} = 90

$a_{12} = 90$

a_{12} = 90

$a_{12} = 90$

a_{12} = 90

$a_{12} = 90$

a_{12} = 90

$a_{12} = 90$

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} 33 & - 15 - 45 & 30 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 33 & - 15 \\ - 45 & 30 \end{array} |$

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} = 33 \cdot 30 - (- 45 \cdot - 15)

$a_{13} = 33 \cdot 30 - (- 45 \cdot - 15)$

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

33

$33$ by

30

$30$ .

a_{13} = 990 - (- 45 \cdot - 15)

$a_{13} = 990 - (- 45 \cdot - 15)$

Step 2.3.2.2.1.2

Multiply

- (- 45 \cdot - 15)

$- (- 45 \cdot - 15)$ .

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Step 2.3.2.2.1.2.1

Multiply

- 45

$- 45$ by

- 15

$- 15$ .

a_{13} = 990 - 1 \cdot 675

$a_{13} = 990 - 1 \cdot 675$

Step 2.3.2.2.1.2.2

Multiply

- 1

$- 1$ by

675

$675$ .

a_{13} = 990 - 675

$a_{13} = 990 - 675$

a_{13} = 990 - 675

$a_{13} = 990 - 675$

a_{13} = 990 - 675

$a_{13} = 990 - 675$

Step 2.3.2.2.2

Subtract

675

$675$ from

990

$990$ .

a_{13} = 315

$a_{13} = 315$

a_{13} = 315

$a_{13} = 315$

a_{13} = 315

$a_{13} = 315$

a_{13} = 315

$a_{13} = 315$

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} 30 & 24 30 & 15 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 30 & 24 \\ 30 & 15 \end{array} |$

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} = 30 \cdot 15 - 30 \cdot 24

$a_{21} = 30 \cdot 15 - 30 \cdot 24$

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

30

$30$ by

15

$15$ .

a_{21} = 450 - 30 \cdot 24

$a_{21} = 450 - 30 \cdot 24$

Step 2.4.2.2.1.2

Multiply

- 30

$- 30$ by

24

$24$ .

a_{21} = 450 - 720

$a_{21} = 450 - 720$

a_{21} = 450 - 720

$a_{21} = 450 - 720$

Step 2.4.2.2.2

Subtract

720

$720$ from

450

$450$ .

a_{21} = - 270

$a_{21} = - 270$

Step 2.5

Calculate the minor for element

$a_{22}$ .

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Step 2.5.1

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

$| \begin{array}{c} - 63 & 24 \\ - 45 & 15 \end{array} |$

Step 2.5.2

Evaluate the determinant.

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Step 2.5.2.1

The determinant of a

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

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

$a_{22} = - 63 \cdot 15 - (- 45 \cdot 24)$

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

$- 63$ by

$15$ .

$a_{22} = - 945 - (- 45 \cdot 24)$

Step 2.5.2.2.1.2

Multiply

$- (- 45 \cdot 24)$ .

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Step 2.5.2.2.1.2.1

Multiply

$- 45$ by

$24$ .

$a_{22} = - 945 - - 1080$

Step 2.5.2.2.1.2.2

Multiply

$- 1$ by

$- 1080$ .

$a_{22} = - 945 + 1080$

Step 2.5.2.2.2

Add

$- 945$ and

$1080$ .

$a_{22} = 135$

Step 2.6

Calculate the minor for element

$a_{23}$ .

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Step 2.6.1

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

$| \begin{array}{c} - 63 & 30 \\ - 45 & 30 \end{array} |$

Step 2.6.2

Evaluate the determinant.

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Step 2.6.2.1

The determinant of a

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

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

$a_{23} = - 63 \cdot 30 - (- 45 \cdot 30)$

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

$- 63$ by

$30$ .

$a_{23} = - 1890 - (- 45 \cdot 30)$

Step 2.6.2.2.1.2

Multiply

$- (- 45 \cdot 30)$ .

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Step 2.6.2.2.1.2.1

Multiply

$- 45$ by

$30$ .

$a_{23} = - 1890 - - 1350$

Step 2.6.2.2.1.2.2

Multiply

$- 1$ by

$- 1350$ .

$a_{23} = - 1890 + 1350$

Step 2.6.2.2.2

Add

$- 1890$ and

$1350$ .

$a_{23} = - 540$

Step 2.7

Calculate the minor for element

$a_{31}$ .

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Step 2.7.1

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

$| \begin{array}{c} 30 & 24 \\ - 15 & - 9 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

The determinant of a

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

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

$a_{31} = 30 \cdot - 9 - (- 15 \cdot 24)$

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

$30$ by

$- 9$ .

$a_{31} = - 270 - (- 15 \cdot 24)$

Step 2.7.2.2.1.2

Multiply

$- (- 15 \cdot 24)$ .

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Step 2.7.2.2.1.2.1

Multiply

$- 15$ by

$24$ .

$a_{31} = - 270 - - 360$

Step 2.7.2.2.1.2.2

Multiply

$- 1$ by

$- 360$ .

$a_{31} = - 270 + 360$

Step 2.7.2.2.2

Add

$- 270$ and

$360$ .

$a_{31} = 90$

Step 2.8

Calculate the minor for element

$a_{32}$ .

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Step 2.8.1

The minor for

$a_{32}$ is the determinant with row

$3$ and column

$2$ deleted.

$| \begin{array}{c} - 63 & 24 \\ 33 & - 9 \end{array} |$

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

The determinant of a

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

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

$a_{32} = - 63 \cdot - 9 - 33 \cdot 24$

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

$- 63$ by

$- 9$ .

$a_{32} = 567 - 33 \cdot 24$

Step 2.8.2.2.1.2

Multiply

$- 33$ by

$24$ .

$a_{32} = 567 - 792$

Step 2.8.2.2.2

Subtract

$792$ from

$567$ .

$a_{32} = - 225$

Step 2.9

Calculate the minor for element

$a_{33}$ .

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Step 2.9.1

The minor for

$a_{33}$ is the determinant with row

$3$ and column

$3$ deleted.

$| \begin{array}{c} - 63 & 30 \\ 33 & - 15 \end{array} |$

Step 2.9.2

Evaluate the determinant.

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Step 2.9.2.1

The determinant of a

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

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

$a_{33} = - 63 \cdot - 15 - 33 \cdot 30$

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

$- 63$ by

$- 15$ .

$a_{33} = 945 - 33 \cdot 30$

Step 2.9.2.2.1.2

Multiply

$- 33$ by

$30$ .

$a_{33} = 945 - 990$

Step 2.9.2.2.2

Subtract

$990$ from

$945$ .

$a_{33} = - 45$

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.

$[\begin{array}{c} 45 & - 90 & 315 \\ 270 & 135 & 540 \\ 90 & 225 & - 45 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 45 & 270 & 90 \\ - 90 & 135 & 225 \\ 315 & 540 & - 45 \end{array}]$