Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 2 & 6 0 & - 4 & - 12 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 2 & 6 \\ 0 & - 4 & - 12 \end{array}]$

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} 2 & 6 - 4 & - 12 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 6 \\ - 4 & - 12 \end{array} |$

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} = 2 \cdot - 12 - (- 4 \cdot 6)

$a_{11} = 2 \cdot - 12 - (- 4 \cdot 6)$

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

2

$2$ by

- 12

$- 12$ .

a_{11} = - 24 - (- 4 \cdot 6)

$a_{11} = - 24 - (- 4 \cdot 6)$

Step 2.1.2.2.1.2

Multiply

- (- 4 \cdot 6)

$- (- 4 \cdot 6)$ .

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Step 2.1.2.2.1.2.1

Multiply

- 4

$- 4$ by

6

$6$ .

a_{11} = - 24 - - 24

$a_{11} = - 24 - - 24$

Step 2.1.2.2.1.2.2

Multiply

- 1

$- 1$ by

- 24

$- 24$ .

a_{11} = - 24 + 24

$a_{11} = - 24 + 24$

a_{11} = - 24 + 24

$a_{11} = - 24 + 24$

a_{11} = - 24 + 24

$a_{11} = - 24 + 24$

Step 2.1.2.2.2

Add

- 24

$- 24$ and

24

$24$ .

a_{11} = 0

$a_{11} = 0$

a_{11} = 0

$a_{11} = 0$

a_{11} = 0

$a_{11} = 0$

a_{11} = 0

$a_{11} = 0$

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} 0 & 6 0 & - 12 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 0 & 6 \\ 0 & - 12 \end{array} |$

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} = 0 \cdot - 12 + 0 \cdot 6

$a_{12} = 0 \cdot - 12 + 0 \cdot 6$

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

0

$0$ by

- 12

$- 12$ .

a_{12} = 0 + 0 \cdot 6

$a_{12} = 0 + 0 \cdot 6$

Step 2.2.2.2.1.2

Multiply

0

$0$ by

6

$6$ .

a_{12} = 0 + 0

$a_{12} = 0 + 0$

a_{12} = 0 + 0

$a_{12} = 0 + 0$

Step 2.2.2.2.2

Add

0

$0$ and

0

$0$ .

a_{12} = 0

$a_{12} = 0$

a_{12} = 0

$a_{12} = 0$

a_{12} = 0

$a_{12} = 0$

a_{12} = 0

$a_{12} = 0$

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} 0 & 2 0 & - 4 \end{matrix} ∣ ∣ ∣

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

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

$a_{13} = 0 \cdot - 4 + 0 \cdot 2$

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

0

$0$ by

- 4

$- 4$ .

a_{13} = 0 + 0 \cdot 2

$a_{13} = 0 + 0 \cdot 2$

Step 2.3.2.2.1.2

Multiply

0

$0$ by

2

$2$ .

a_{13} = 0 + 0

$a_{13} = 0 + 0$

a_{13} = 0 + 0

$a_{13} = 0 + 0$

Step 2.3.2.2.2

Add

0

$0$ and

0

$0$ .

a_{13} = 0

$a_{13} = 0$

a_{13} = 0

$a_{13} = 0$

a_{13} = 0

$a_{13} = 0$

a_{13} = 0

$a_{13} = 0$

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 & 0 - 4 & - 12 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 0 & 0 \\ - 4 & - 12 \end{array} |$

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

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

$a_{21} = 0 \cdot - 12 - (- 4 \cdot 0)$

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

$0$ by

$- 12$ .

$a_{21} = 0 - (- 4 \cdot 0)$

Step 2.4.2.2.1.2

Multiply

$- (- 4 \cdot 0)$ .

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Step 2.4.2.2.1.2.1

Multiply

$- 4$ by

$0$ .

$a_{21} = 0 - 0$

Step 2.4.2.2.1.2.2

Multiply

$- 1$ by

$0$ .

$a_{21} = 0 + 0$

Step 2.4.2.2.2

Add

$0$ and

$0$ .

$a_{21} = 0$

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.

$| \begin{array}{c} 1 & 0 \\ 0 & - 12 \end{array} |$

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

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

$a_{22} = 1 \cdot - 12 + 0 \cdot 0$

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

$- 12$ by

$1$ .

$a_{22} = - 12 + 0 \cdot 0$

Step 2.5.2.2.1.2

Multiply

$0$ by

$0$ .

$a_{22} = - 12 + 0$

Step 2.5.2.2.2

Add

$- 12$ and

$0$ .

$a_{22} = - 12$

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.

$| \begin{array}{c} 1 & 0 \\ 0 & - 4 \end{array} |$

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

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

$a_{23} = 1 \cdot - 4 + 0 \cdot 0$

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

$- 4$ by

$1$ .

$a_{23} = - 4 + 0 \cdot 0$

Step 2.6.2.2.1.2

Multiply

$0$ by

$0$ .

$a_{23} = - 4 + 0$

Step 2.6.2.2.2

Add

$- 4$ and

$0$ .

$a_{23} = - 4$

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.

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

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

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

$a_{31} = 0 \cdot 6 - 2 \cdot 0$

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

$0$ by

$6$ .

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

Step 2.7.2.2.1.2

Multiply

$- 2$ by

$0$ .

$a_{31} = 0 + 0$

Step 2.7.2.2.2

Add

$0$ and

$0$ .

$a_{31} = 0$

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.

$| \begin{array}{c} 1 & 0 \\ 0 & 6 \end{array} |$

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

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

$a_{32} = 1 \cdot 6 + 0 \cdot 0$

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

$6$ by

$1$ .

$a_{32} = 6 + 0 \cdot 0$

Step 2.8.2.2.1.2

Multiply

$0$ by

$0$ .

$a_{32} = 6 + 0$

Step 2.8.2.2.2

Add

$6$ and

$0$ .

$a_{32} = 6$

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.

$| \begin{array}{c} 1 & 0 \\ 0 & 2 \end{array} |$

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

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

$a_{33} = 1 \cdot 2 + 0 \cdot 0$

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

$2$ by

$1$ .

$a_{33} = 2 + 0 \cdot 0$

Step 2.9.2.2.1.2

Multiply

$0$ by

$0$ .

$a_{33} = 2 + 0$

Step 2.9.2.2.2

Add

$2$ and

$0$ .

$a_{33} = 2$

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.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & - 12 & 4 \\ 0 & - 6 & 2 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & - 12 & - 6 \\ 0 & 4 & 2 \end{array}]$