Linear Algebra Examples

$B = [\begin{array}{c} 1 & 2 & - 1 \\ 5 & 4 & 3 \\ 2 & - 4 & 8 \end{array}]$

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 $b_{11}$ .

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Step 2.1.1

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

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

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

$b_{11} = 4 \cdot 8 - (- 4 \cdot 3)$

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 $4$ by $8$ .

$b_{11} = 32 - (- 4 \cdot 3)$

Step 2.1.2.2.1.2

Multiply $- (- 4 \cdot 3)$ .

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Step 2.1.2.2.1.2.1

Multiply $- 4$ by $3$ .

$b_{11} = 32 - - 12$

Step 2.1.2.2.1.2.2

Multiply $- 1$ by $- 12$ .

$b_{11} = 32 + 12$

Step 2.1.2.2.2

Add $32$ and $12$ .

$b_{11} = 44$

Step 2.2

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

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Step 2.2.1

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

$| \begin{array}{c} 5 & 3 \\ 2 & 8 \end{array} |$

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

$b_{12} = 5 \cdot 8 - 2 \cdot 3$

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 $8$ .

$b_{12} = 40 - 2 \cdot 3$

Step 2.2.2.2.1.2

Multiply $- 2$ by $3$ .

$b_{12} = 40 - 6$

Step 2.2.2.2.2

Subtract $6$ from $40$ .

$b_{12} = 34$

Step 2.3

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

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Step 2.3.1

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

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

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

$b_{13} = 5 \cdot - 4 - 2 \cdot 4$

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 $- 4$ .

$b_{13} = - 20 - 2 \cdot 4$

Step 2.3.2.2.1.2

Multiply $- 2$ by $4$ .

$b_{13} = - 20 - 8$

Step 2.3.2.2.2

Subtract $8$ from $- 20$ .

$b_{13} = - 28$

Step 2.4

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

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Step 2.4.1

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

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

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

$b_{21} = 2 \cdot 8 - (- 4 \cdot - 1)$

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 $2$ by $8$ .

$b_{21} = 16 - (- 4 \cdot - 1)$

Step 2.4.2.2.1.2

Multiply $- (- 4 \cdot - 1)$ .

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Step 2.4.2.2.1.2.1

Multiply $- 4$ by $- 1$ .

$b_{21} = 16 - 1 \cdot 4$

Step 2.4.2.2.1.2.2

Multiply $- 1$ by $4$ .

$b_{21} = 16 - 4$

Step 2.4.2.2.2

Subtract $4$ from $16$ .

$b_{21} = 12$

Step 2.5

Calculate the minor for element $b_{22}$ .

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Step 2.5.1

The minor for $b_{22}$ is the determinant with row $2$ and column $2$ deleted.

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

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

$b_{22} = 1 \cdot 8 - 2 \cdot - 1$

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 $8$ by $1$ .

$b_{22} = 8 - 2 \cdot - 1$

Step 2.5.2.2.1.2

Multiply $- 2$ by $- 1$ .

$b_{22} = 8 + 2$

Step 2.5.2.2.2

Add $8$ and $2$ .

$b_{22} = 10$

Step 2.6

Calculate the minor for element $b_{23}$ .

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Step 2.6.1

The minor for $b_{23}$ is the determinant with row $2$ and column $3$ deleted.

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

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

$b_{23} = 1 \cdot - 4 - 2 \cdot 2$

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 $1$ .

$b_{23} = - 4 - 2 \cdot 2$

Step 2.6.2.2.1.2

Multiply $- 2$ by $2$ .

$b_{23} = - 4 - 4$

Step 2.6.2.2.2

Subtract $4$ from $- 4$ .

$b_{23} = - 8$

Step 2.7

Calculate the minor for element $b_{31}$ .

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Step 2.7.1

The minor for $b_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 2 & - 1 \\ 4 & 3 \end{array} |$

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

$b_{31} = 2 \cdot 3 - 4 \cdot - 1$

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 $2$ by $3$ .

$b_{31} = 6 - 4 \cdot - 1$

Step 2.7.2.2.1.2

Multiply $- 4$ by $- 1$ .

$b_{31} = 6 + 4$

Step 2.7.2.2.2

Add $6$ and $4$ .

$b_{31} = 10$

Step 2.8

Calculate the minor for element $b_{32}$ .

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Step 2.8.1

The minor for $b_{32}$ is the determinant with row $3$ and column $2$ deleted.

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

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

$b_{32} = 1 \cdot 3 - 5 \cdot - 1$

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 $3$ by $1$ .

$b_{32} = 3 - 5 \cdot - 1$

Step 2.8.2.2.1.2

Multiply $- 5$ by $- 1$ .

$b_{32} = 3 + 5$

Step 2.8.2.2.2

Add $3$ and $5$ .

$b_{32} = 8$

Step 2.9

Calculate the minor for element $b_{33}$ .

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Step 2.9.1

The minor for $b_{33}$ is the determinant with row $3$ and column $3$ deleted.

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

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

$b_{33} = 1 \cdot 4 - 5 \cdot 2$

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 $4$ by $1$ .

$b_{33} = 4 - 5 \cdot 2$

Step 2.9.2.2.1.2

Multiply $- 5$ by $2$ .

$b_{33} = 4 - 10$

Step 2.9.2.2.2

Subtract $10$ from $4$ .

$b_{33} = - 6$

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.

$[\begin{array}{c} 44 & - 34 & - 28 \\ - 12 & 10 & 8 \\ 10 & - 8 & - 6 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 44 & - 12 & 10 \\ - 34 & 10 & - 8 \\ - 28 & 8 & - 6 \end{array}]$

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