Linear Algebra Examples

$[\begin{array}{c} 9 & 8 & 7 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{array}]$

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.

$| \begin{array}{c} 5 & 6 \\ 2 & 3 \end{array} |$

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

$a_{11} = 5 \cdot 3 - 2 \cdot 6$

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 $5$ by $3$ .

$a_{11} = 15 - 2 \cdot 6$

Step 2.1.2.2.1.2

Multiply $- 2$ by $6$ .

$a_{11} = 15 - 12$

Step 2.1.2.2.2

Subtract $12$ from $15$ .

$a_{11} = 3$

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.

$| \begin{array}{c} 4 & 6 \\ 1 & 3 \end{array} |$

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

$a_{12} = 4 \cdot 3 - 1 \cdot 6$

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 $4$ by $3$ .

$a_{12} = 12 - 1 \cdot 6$

Step 2.2.2.2.1.2

Multiply $- 1$ by $6$ .

$a_{12} = 12 - 6$

Step 2.2.2.2.2

Subtract $6$ from $12$ .

$a_{12} = 6$

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.

$| \begin{array}{c} 4 & 5 \\ 1 & 2 \end{array} |$

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

$a_{13} = 4 \cdot 2 - 1 \cdot 5$

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 $4$ by $2$ .

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

Step 2.3.2.2.1.2

Multiply $- 1$ by $5$ .

$a_{13} = 8 - 5$

Step 2.3.2.2.2

Subtract $5$ from $8$ .

$a_{13} = 3$

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.

$| \begin{array}{c} 8 & 7 \\ 2 & 3 \end{array} |$

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

$a_{21} = 8 \cdot 3 - 2 \cdot 7$

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 $8$ by $3$ .

$a_{21} = 24 - 2 \cdot 7$

Step 2.4.2.2.1.2

Multiply $- 2$ by $7$ .

$a_{21} = 24 - 14$

Step 2.4.2.2.2

Subtract $14$ from $24$ .

$a_{21} = 10$

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.

$| \begin{array}{c} 9 & 7 \\ 1 & 3 \end{array} |$

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

$a_{22} = 9 \cdot 3 - 1 \cdot 7$

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 $3$ .

$a_{22} = 27 - 1 \cdot 7$

Step 2.5.2.2.1.2

Multiply $- 1$ by $7$ .

$a_{22} = 27 - 7$

Step 2.5.2.2.2

Subtract $7$ from $27$ .

$a_{22} = 20$

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.

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

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

$a_{23} = 9 \cdot 2 - 1 \cdot 8$

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 $9$ by $2$ .

$a_{23} = 18 - 1 \cdot 8$

Step 2.6.2.2.1.2

Multiply $- 1$ by $8$ .

$a_{23} = 18 - 8$

Step 2.6.2.2.2

Subtract $8$ from $18$ .

$a_{23} = 10$

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.

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

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

$a_{31} = 8 \cdot 6 - 5 \cdot 7$

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 $8$ by $6$ .

$a_{31} = 48 - 5 \cdot 7$

Step 2.7.2.2.1.2

Multiply $- 5$ by $7$ .

$a_{31} = 48 - 35$

Step 2.7.2.2.2

Subtract $35$ from $48$ .

$a_{31} = 13$

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.

$| \begin{array}{c} 9 & 7 \\ 4 & 6 \end{array} |$

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

$a_{32} = 9 \cdot 6 - 4 \cdot 7$

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 $9$ by $6$ .

$a_{32} = 54 - 4 \cdot 7$

Step 2.8.2.2.1.2

Multiply $- 4$ by $7$ .

$a_{32} = 54 - 28$

Step 2.8.2.2.2

Subtract $28$ from $54$ .

$a_{32} = 26$

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.

$| \begin{array}{c} 9 & 8 \\ 4 & 5 \end{array} |$

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

$a_{33} = 9 \cdot 5 - 4 \cdot 8$

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 $9$ by $5$ .

$a_{33} = 45 - 4 \cdot 8$

Step 2.9.2.2.1.2

Multiply $- 4$ by $8$ .

$a_{33} = 45 - 32$

Step 2.9.2.2.2

Subtract $32$ from $45$ .

$a_{33} = 13$

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.

$[\begin{array}{c} 3 & - 6 & 3 \\ - 10 & 20 & - 10 \\ 13 & - 26 & 13 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 3 & - 10 & 13 \\ - 6 & 20 & - 26 \\ 3 & - 10 & 13 \end{array}]$

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