Linear Algebra Examples

$[\begin{array}{c} 1 & 3 & 4 \\ - 2 & - 6 & - 8 \\ 1 & 4 & 9 \end{array}]$

Step 1

Consider the corresponding sign chart.

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

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 $a_{11}$ .

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Step 2.1.1

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 6 & - 8 \\ 4 & 9 \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{11} = - 6 \cdot 9 - 4 \cdot - 8$

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 $- 6$ by $9$ .

$a_{11} = - 54 - 4 \cdot - 8$

Step 2.1.2.2.1.2

Multiply $- 4$ by $- 8$ .

$a_{11} = - 54 + 32$

Step 2.1.2.2.2

Add $- 54$ and $32$ .

$a_{11} = - 22$

Step 2.2

Calculate the minor for element $a_{12}$ .

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Step 2.2.1

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} - 2 & - 8 \\ 1 & 9 \end{array} |$

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{12} = - 2 \cdot 9 - 1 \cdot - 8$

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 $- 2$ by $9$ .

$a_{12} = - 18 - 1 \cdot - 8$

Step 2.2.2.2.1.2

Multiply $- 1$ by $- 8$ .

$a_{12} = - 18 + 8$

Step 2.2.2.2.2

Add $- 18$ and $8$ .

$a_{12} = - 10$

Step 2.3

Calculate the minor for element $a_{13}$ .

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Step 2.3.1

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} - 2 & - 6 \\ 1 & 4 \end{array} |$

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{13} = - 2 \cdot 4 - 1 \cdot - 6$

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 $- 2$ by $4$ .

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

Step 2.3.2.2.1.2

Multiply $- 1$ by $- 6$ .

$a_{13} = - 8 + 6$

Step 2.3.2.2.2

Add $- 8$ and $6$ .

$a_{13} = - 2$

Step 2.4

Calculate the minor for element $a_{21}$ .

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Step 2.4.1

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 3 & 4 \\ 4 & 9 \end{array} |$

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{21} = 3 \cdot 9 - 4 \cdot 4$

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 $3$ by $9$ .

$a_{21} = 27 - 4 \cdot 4$

Step 2.4.2.2.1.2

Multiply $- 4$ by $4$ .

$a_{21} = 27 - 16$

Step 2.4.2.2.2

Subtract $16$ from $27$ .

$a_{21} = 11$

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.

$| \begin{array}{c} 1 & 4 \\ 1 & 9 \end{array} |$

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

$a_{22} = 1 \cdot 9 - 1 \cdot 4$

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 $9$ by $1$ .

$a_{22} = 9 - 1 \cdot 4$

Step 2.5.2.2.1.2

Multiply $- 1$ by $4$ .

$a_{22} = 9 - 4$

Step 2.5.2.2.2

Subtract $4$ from $9$ .

$a_{22} = 5$

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.

$| \begin{array}{c} 1 & 3 \\ 1 & 4 \end{array} |$

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

$a_{23} = 1 \cdot 4 - 1 \cdot 3$

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 $1$ .

$a_{23} = 4 - 1 \cdot 3$

Step 2.6.2.2.1.2

Multiply $- 1$ by $3$ .

$a_{23} = 4 - 3$

Step 2.6.2.2.2

Subtract $3$ from $4$ .

$a_{23} = 1$

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.

$| \begin{array}{c} 3 & 4 \\ - 6 & - 8 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{31} = 3 \cdot - 8 - (- 6 \cdot 4)$

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 $3$ by $- 8$ .

$a_{31} = - 24 - (- 6 \cdot 4)$

Step 2.7.2.2.1.2

Multiply $- (- 6 \cdot 4)$ .

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Step 2.7.2.2.1.2.1

Multiply $- 6$ by $4$ .

$a_{31} = - 24 - - 24$

Step 2.7.2.2.1.2.2

Multiply $- 1$ by $- 24$ .

$a_{31} = - 24 + 24$

Step 2.7.2.2.2

Add $- 24$ and $24$ .

$a_{31} = 0$

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.

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

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{32} = 1 \cdot - 8 - (- 2 \cdot 4)$

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 $- 8$ by $1$ .

$a_{32} = - 8 - (- 2 \cdot 4)$

Step 2.8.2.2.1.2

Multiply $- (- 2 \cdot 4)$ .

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Step 2.8.2.2.1.2.1

Multiply $- 2$ by $4$ .

$a_{32} = - 8 - - 8$

Step 2.8.2.2.1.2.2

Multiply $- 1$ by $- 8$ .

$a_{32} = - 8 + 8$

Step 2.8.2.2.2

Add $- 8$ and $8$ .

$a_{32} = 0$

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.

$| \begin{array}{c} 1 & 3 \\ - 2 & - 6 \end{array} |$

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

$a_{33} = 1 \cdot - 6 - (- 2 \cdot 3)$

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 $- 6$ by $1$ .

$a_{33} = - 6 - (- 2 \cdot 3)$

Step 2.9.2.2.1.2

Multiply $- (- 2 \cdot 3)$ .

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Step 2.9.2.2.1.2.1

Multiply $- 2$ by $3$ .

$a_{33} = - 6 - - 6$

Step 2.9.2.2.1.2.2

Multiply $- 1$ by $- 6$ .

$a_{33} = - 6 + 6$

Step 2.9.2.2.2

Add $- 6$ and $6$ .

$a_{33} = 0$

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.

$[\begin{array}{c} - 22 & 10 & - 2 \\ - 11 & 5 & - 1 \\ 0 & 0 & 0 \end{array}]$