Linear Algebra Examples

$[\begin{array}{c} 6 & 4 & 2 \\ 5 & 1 & 8 \\ 6 & 8 & 7 \end{array}]$

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.

$| \begin{array}{c} 1 & 8 \\ 8 & 7 \end{array} |$

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

$a_{11} = 1 \cdot 7 - 8 \cdot 8$

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 $7$ by $1$ .

$a_{11} = 7 - 8 \cdot 8$

Step 2.1.2.2.1.2

Multiply $- 8$ by $8$ .

$a_{11} = 7 - 64$

Step 2.1.2.2.2

Subtract $64$ from $7$ .

$a_{11} = - 57$

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.

$| \begin{array}{c} 5 & 8 \\ 6 & 7 \end{array} |$

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

$a_{12} = 5 \cdot 7 - 6 \cdot 8$

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 $5$ by $7$ .

$a_{12} = 35 - 6 \cdot 8$

Step 2.2.2.2.1.2

Multiply $- 6$ by $8$ .

$a_{12} = 35 - 48$

Step 2.2.2.2.2

Subtract $48$ from $35$ .

$a_{12} = - 13$

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.

$| \begin{array}{c} 5 & 1 \\ 6 & 8 \end{array} |$

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

$a_{13} = 5 \cdot 8 - 6 \cdot 1$

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 $5$ by $8$ .

$a_{13} = 40 - 6 \cdot 1$

Step 2.3.2.2.1.2

Multiply $- 6$ by $1$ .

$a_{13} = 40 - 6$

Step 2.3.2.2.2

Subtract $6$ from $40$ .

$a_{13} = 34$

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.

$| \begin{array}{c} 4 & 2 \\ 8 & 7 \end{array} |$

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

$a_{21} = 4 \cdot 7 - 8 \cdot 2$

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 $4$ by $7$ .

$a_{21} = 28 - 8 \cdot 2$

Step 2.4.2.2.1.2

Multiply $- 8$ by $2$ .

$a_{21} = 28 - 16$

Step 2.4.2.2.2

Subtract $16$ from $28$ .

$a_{21} = 12$

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.

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

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

$a_{22} = 6 \cdot 7 - 6 \cdot 2$

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 $6$ by $7$ .

$a_{22} = 42 - 6 \cdot 2$

Step 2.5.2.2.1.2

Multiply $- 6$ by $2$ .

$a_{22} = 42 - 12$

Step 2.5.2.2.2

Subtract $12$ from $42$ .

$a_{22} = 30$

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.

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

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

$a_{23} = 6 \cdot 8 - 6 \cdot 4$

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 $6$ by $8$ .

$a_{23} = 48 - 6 \cdot 4$

Step 2.6.2.2.1.2

Multiply $- 6$ by $4$ .

$a_{23} = 48 - 24$

Step 2.6.2.2.2

Subtract $24$ from $48$ .

$a_{23} = 24$

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.

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

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

$a_{31} = 4 \cdot 8 - 1 \cdot 2$

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 $4$ by $8$ .

$a_{31} = 32 - 1 \cdot 2$

Step 2.7.2.2.1.2

Multiply $- 1$ by $2$ .

$a_{31} = 32 - 2$

Step 2.7.2.2.2

Subtract $2$ from $32$ .

$a_{31} = 30$

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.

$| \begin{array}{c} 6 & 2 \\ 5 & 8 \end{array} |$

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

$a_{32} = 6 \cdot 8 - 5 \cdot 2$

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 $6$ by $8$ .

$a_{32} = 48 - 5 \cdot 2$

Step 2.8.2.2.1.2

Multiply $- 5$ by $2$ .

$a_{32} = 48 - 10$

Step 2.8.2.2.2

Subtract $10$ from $48$ .

$a_{32} = 38$

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.

$| \begin{array}{c} 6 & 4 \\ 5 & 1 \end{array} |$

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

$a_{33} = 6 \cdot 1 - 5 \cdot 4$

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 $1$ .

$a_{33} = 6 - 5 \cdot 4$

Step 2.9.2.2.1.2

Multiply $- 5$ by $4$ .

$a_{33} = 6 - 20$

Step 2.9.2.2.2

Subtract $20$ from $6$ .

$a_{33} = - 14$

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.

$[\begin{array}{c} - 57 & 13 & 34 \\ - 12 & 30 & - 24 \\ 30 & - 38 & - 14 \end{array}]$