Linear Algebra Examples

$[\begin{array}{c} 2 & - 3 & 3 \\ 2 & 1 & 1 \\ 1 & 5 & - 1 \end{array}]$

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.

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

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

$a_{11} = 1 \cdot - 1 - 5 \cdot 1$

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 $- 1$ by $1$ .

$a_{11} = - 1 - 5 \cdot 1$

Step 2.1.2.2.1.2

Multiply $- 5$ by $1$ .

$a_{11} = - 1 - 5$

Step 2.1.2.2.2

Subtract $5$ from $- 1$ .

$a_{11} = - 6$

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.

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

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

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

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 $- 1$ .

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

Step 2.2.2.2.1.2

Multiply $- 1$ by $1$ .

$a_{12} = - 2 - 1$

Step 2.2.2.2.2

Subtract $1$ from $- 2$ .

$a_{12} = - 3$

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.

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

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

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

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 $2$ by $5$ .

$a_{13} = 10 - 1 \cdot 1$

Step 2.3.2.2.1.2

Multiply $- 1$ by $1$ .

$a_{13} = 10 - 1$

Step 2.3.2.2.2

Subtract $1$ from $10$ .

$a_{13} = 9$

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.

$| \begin{array}{c} - 3 & 3 \\ 5 & - 1 \end{array} |$

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

$a_{21} = - 3 \cdot - 1 - 5 \cdot 3$

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 $- 1$ .

$a_{21} = 3 - 5 \cdot 3$

Step 2.4.2.2.1.2

Multiply $- 5$ by $3$ .

$a_{21} = 3 - 15$

Step 2.4.2.2.2

Subtract $15$ from $3$ .

$a_{21} = - 12$

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.

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

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

$a_{22} = 2 \cdot - 1 - 1 \cdot 3$

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 $2$ by $- 1$ .

$a_{22} = - 2 - 1 \cdot 3$

Step 2.5.2.2.1.2

Multiply $- 1$ by $3$ .

$a_{22} = - 2 - 3$

Step 2.5.2.2.2

Subtract $3$ from $- 2$ .

$a_{22} = - 5$

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.

$| \begin{array}{c} 2 & - 3 \\ 1 & 5 \end{array} |$

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

$a_{23} = 2 \cdot 5 - 1 \cdot - 3$

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 $2$ by $5$ .

$a_{23} = 10 - 1 \cdot - 3$

Step 2.6.2.2.1.2

Multiply $- 1$ by $- 3$ .

$a_{23} = 10 + 3$

Step 2.6.2.2.2

Add $10$ and $3$ .

$a_{23} = 13$

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.

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

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

$a_{31} = - 3 \cdot 1 - 1 \cdot 3$

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 $1$ .

$a_{31} = - 3 - 1 \cdot 3$

Step 2.7.2.2.1.2

Multiply $- 1$ by $3$ .

$a_{31} = - 3 - 3$

Step 2.7.2.2.2

Subtract $3$ from $- 3$ .

$a_{31} = - 6$

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.

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

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

$a_{32} = 2 \cdot 1 - 2 \cdot 3$

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 $2$ by $1$ .

$a_{32} = 2 - 2 \cdot 3$

Step 2.8.2.2.1.2

Multiply $- 2$ by $3$ .

$a_{32} = 2 - 6$

Step 2.8.2.2.2

Subtract $6$ from $2$ .

$a_{32} = - 4$

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.

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

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

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

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 $2$ by $1$ .

$a_{33} = 2 - 2 \cdot - 3$

Step 2.9.2.2.1.2

Multiply $- 2$ by $- 3$ .

$a_{33} = 2 + 6$

Step 2.9.2.2.2

Add $2$ and $6$ .

$a_{33} = 8$

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.

$[\begin{array}{c} - 6 & 3 & 9 \\ 12 & - 5 & - 13 \\ - 6 & 4 & 8 \end{array}]$