Linear Algebra Examples

$[\begin{array}{c} 17 & 22 & 4 \\ 2 & 9 & 11 \\ 4 & 6 & 13 \end{array}]$

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.

$| \begin{array}{c} 9 & 11 \\ 6 & 13 \end{array} |$

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

$a_{11} = 9 \cdot 13 - 6 \cdot 11$

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 $9$ by $13$ .

$a_{11} = 117 - 6 \cdot 11$

Step 2.1.2.2.1.2

Multiply $- 6$ by $11$ .

$a_{11} = 117 - 66$

Step 2.1.2.2.2

Subtract $66$ from $117$ .

$a_{11} = 51$

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.

$| \begin{array}{c} 2 & 11 \\ 4 & 13 \end{array} |$

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

$a_{12} = 2 \cdot 13 - 4 \cdot 11$

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 $13$ .

$a_{12} = 26 - 4 \cdot 11$

Step 2.2.2.2.1.2

Multiply $- 4$ by $11$ .

$a_{12} = 26 - 44$

Step 2.2.2.2.2

Subtract $44$ from $26$ .

$a_{12} = - 18$

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.

$| \begin{array}{c} 2 & 9 \\ 4 & 6 \end{array} |$

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

$a_{13} = 2 \cdot 6 - 4 \cdot 9$

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 $6$ .

$a_{13} = 12 - 4 \cdot 9$

Step 2.3.2.2.1.2

Multiply $- 4$ by $9$ .

$a_{13} = 12 - 36$

Step 2.3.2.2.2

Subtract $36$ from $12$ .

$a_{13} = - 24$

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.

$| \begin{array}{c} 22 & 4 \\ 6 & 13 \end{array} |$

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

$a_{21} = 22 \cdot 13 - 6 \cdot 4$

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 $22$ by $13$ .

$a_{21} = 286 - 6 \cdot 4$

Step 2.4.2.2.1.2

Multiply $- 6$ by $4$ .

$a_{21} = 286 - 24$

Step 2.4.2.2.2

Subtract $24$ from $286$ .

$a_{21} = 262$

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.

$| \begin{array}{c} 17 & 4 \\ 4 & 13 \end{array} |$

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

$a_{22} = 17 \cdot 13 - 4 \cdot 4$

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 $17$ by $13$ .

$a_{22} = 221 - 4 \cdot 4$

Step 2.5.2.2.1.2

Multiply $- 4$ by $4$ .

$a_{22} = 221 - 16$

Step 2.5.2.2.2

Subtract $16$ from $221$ .

$a_{22} = 205$

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.

$| \begin{array}{c} 17 & 22 \\ 4 & 6 \end{array} |$

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

$a_{23} = 17 \cdot 6 - 4 \cdot 22$

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 $17$ by $6$ .

$a_{23} = 102 - 4 \cdot 22$

Step 2.6.2.2.1.2

Multiply $- 4$ by $22$ .

$a_{23} = 102 - 88$

Step 2.6.2.2.2

Subtract $88$ from $102$ .

$a_{23} = 14$

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.

$| \begin{array}{c} 22 & 4 \\ 9 & 11 \end{array} |$

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

$a_{31} = 22 \cdot 11 - 9 \cdot 4$

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 $22$ by $11$ .

$a_{31} = 242 - 9 \cdot 4$

Step 2.7.2.2.1.2

Multiply $- 9$ by $4$ .

$a_{31} = 242 - 36$

Step 2.7.2.2.2

Subtract $36$ from $242$ .

$a_{31} = 206$

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.

$| \begin{array}{c} 17 & 4 \\ 2 & 11 \end{array} |$

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

$a_{32} = 17 \cdot 11 - 2 \cdot 4$

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 $17$ by $11$ .

$a_{32} = 187 - 2 \cdot 4$

Step 2.8.2.2.1.2

Multiply $- 2$ by $4$ .

$a_{32} = 187 - 8$

Step 2.8.2.2.2

Subtract $8$ from $187$ .

$a_{32} = 179$

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.

$| \begin{array}{c} 17 & 22 \\ 2 & 9 \end{array} |$

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

$a_{33} = 17 \cdot 9 - 2 \cdot 22$

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 $17$ by $9$ .

$a_{33} = 153 - 2 \cdot 22$

Step 2.9.2.2.1.2

Multiply $- 2$ by $22$ .

$a_{33} = 153 - 44$

Step 2.9.2.2.2

Subtract $44$ from $153$ .

$a_{33} = 109$

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.

$[\begin{array}{c} 51 & 18 & - 24 \\ - 262 & 205 & - 14 \\ 206 & - 179 & 109 \end{array}]$