Linear Algebra Examples

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

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

$a_{11}$ .

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Step 2.1.1

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

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

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

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

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

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

$8$ .

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

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

$3$ .

$a_{11} = 32 - - 12$

Step 2.1.2.2.1.2.2

Multiply

$- 1$ by

$- 12$ .

$a_{11} = 32 + 12$

Step 2.1.2.2.2

Add

$32$ and

$12$ .

$a_{11} = 44$

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.

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

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

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

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

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

$8$ .

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

Step 2.2.2.2.1.2

Multiply

$- 2$ by

$3$ .

$a_{12} = 40 - 6$

Step 2.2.2.2.2

Subtract

$6$ from

$40$ .

$a_{12} = 34$

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.

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

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

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

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

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

$- 4$ .

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

Step 2.3.2.2.1.2

Multiply

$- 2$ by

$4$ .

$a_{13} = - 20 - 8$

Step 2.3.2.2.2

Subtract

$8$ from

$- 20$ .

$a_{13} = - 28$

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.

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

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

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

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

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

$8$ .

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

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

$- 1$ .

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

Step 2.4.2.2.1.2.2

Multiply

$- 1$ by

$4$ .

$a_{21} = 16 - 4$

Step 2.4.2.2.2

Subtract

$4$ from

$16$ .

$a_{21} = 12$

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.

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

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

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

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

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

$1$ .

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

Step 2.5.2.2.1.2

Multiply

$- 2$ by

$- 1$ .

$a_{22} = 8 + 2$

Step 2.5.2.2.2

Add

$8$ and

$2$ .

$a_{22} = 10$

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.

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

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

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

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

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

$1$ .

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

Step 2.6.2.2.1.2

Multiply

$- 2$ by

$2$ .

$a_{23} = - 4 - 4$

Step 2.6.2.2.2

Subtract

$4$ from

$- 4$ .

$a_{23} = - 8$

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.

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

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

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

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

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

$3$ .

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

Step 2.7.2.2.1.2

Multiply

$- 4$ by

$- 1$ .

$a_{31} = 6 + 4$

Step 2.7.2.2.2

Add

$6$ and

$4$ .

$a_{31} = 10$

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.

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

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

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

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

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

$1$ .

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

Step 2.8.2.2.1.2

Multiply

$- 5$ by

$- 1$ .

$a_{32} = 3 + 5$

Step 2.8.2.2.2

Add

$3$ and

$5$ .

$a_{32} = 8$

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.

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

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

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

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

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

$1$ .

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

Step 2.9.2.2.1.2

Multiply

$- 5$ by

$2$ .

$a_{33} = 4 - 10$

Step 2.9.2.2.2

Subtract

$10$ from

$4$ .

$a_{33} = - 6$

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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