Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 2 & 3 & 3 4 & 6 & 5 \end{matrix} ⎤ ⎥ ⎦

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

Step 1

Consider the corresponding sign chart.

⎡ ⎢ ⎣ \begin{matrix} + & - & + - & + & - + & - & + \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array}]$

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}

$a_{11}$ .

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Step 2.1.1

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 3 & 3 6 & 5 \end{matrix} ∣ ∣ ∣

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

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

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

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

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

3

$3$ by

5

$5$ .

a_{11} = 15 - 6 \cdot 3

$a_{11} = 15 - 6 \cdot 3$

Step 2.1.2.2.1.2

Multiply

- 6

$- 6$ by

3

$3$ .

a_{11} = 15 - 18

$a_{11} = 15 - 18$

a_{11} = 15 - 18

$a_{11} = 15 - 18$

Step 2.1.2.2.2

Subtract

18

$18$ from

15

$15$ .

a_{11} = - 3

$a_{11} = - 3$

a_{11} = - 3

$a_{11} = - 3$

a_{11} = - 3

$a_{11} = - 3$

a_{11} = - 3

$a_{11} = - 3$

Step 2.2

Calculate the minor for element

a_{12}

$a_{12}$ .

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Step 2.2.1

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 2 & 3 4 & 5 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 3 \\ 4 & 5 \end{array} |$

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

a_{12} = 2 \cdot 5 - 4 \cdot 3

$a_{12} = 2 \cdot 5 - 4 \cdot 3$

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

2

$2$ by

5

$5$ .

a_{12} = 10 - 4 \cdot 3

$a_{12} = 10 - 4 \cdot 3$

Step 2.2.2.2.1.2

Multiply

- 4

$- 4$ by

3

$3$ .

a_{12} = 10 - 12

$a_{12} = 10 - 12$

a_{12} = 10 - 12

$a_{12} = 10 - 12$

Step 2.2.2.2.2

Subtract

$12$ from

$10$ .

$a_{12} = - 2$

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.

$| \begin{array}{c} 2 & 3 \\ 4 & 6 \end{array} |$

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

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

$a_{13} = 2 \cdot 6 - 4 \cdot 3$

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

$6$ .

$a_{13} = 12 - 4 \cdot 3$

Step 2.3.2.2.1.2

Multiply

$- 4$ by

$3$ .

$a_{13} = 12 - 12$

Step 2.3.2.2.2

Subtract

$12$ from

$12$ .

$a_{13} = 0$

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.

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

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

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

$a_{21} = 2 \cdot 5 - 6 \cdot 3$

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

$2$ by

$5$ .

$a_{21} = 10 - 6 \cdot 3$

Step 2.4.2.2.1.2

Multiply

$- 6$ by

$3$ .

$a_{21} = 10 - 18$

Step 2.4.2.2.2

Subtract

$18$ from

$10$ .

$a_{21} = - 8$

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.

$| \begin{array}{c} 1 & 3 \\ 4 & 5 \end{array} |$

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

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

$a_{22} = 1 \cdot 5 - 4 \cdot 3$

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

$5$ by

$1$ .

$a_{22} = 5 - 4 \cdot 3$

Step 2.5.2.2.1.2

Multiply

$- 4$ by

$3$ .

$a_{22} = 5 - 12$

Step 2.5.2.2.2

Subtract

$12$ from

$5$ .

$a_{22} = - 7$

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.

$| \begin{array}{c} 1 & 2 \\ 4 & 6 \end{array} |$

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

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

$a_{23} = 1 \cdot 6 - 4 \cdot 2$

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

$6$ by

$1$ .

$a_{23} = 6 - 4 \cdot 2$

Step 2.6.2.2.1.2

Multiply

$- 4$ by

$2$ .

$a_{23} = 6 - 8$

Step 2.6.2.2.2

Subtract

$8$ from

$6$ .

$a_{23} = - 2$

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.

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

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

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

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

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

$2$ by

$3$ .

$a_{31} = 6 - 3 \cdot 3$

Step 2.7.2.2.1.2

Multiply

$- 3$ by

$3$ .

$a_{31} = 6 - 9$

Step 2.7.2.2.2

Subtract

$9$ from

$6$ .

$a_{31} = - 3$

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.

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

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

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

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

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

$3$ by

$1$ .

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

Step 2.8.2.2.1.2

Multiply

$- 2$ by

$3$ .

$a_{32} = 3 - 6$

Step 2.8.2.2.2

Subtract

$6$ from

$3$ .

$a_{32} = - 3$

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.

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

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

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

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

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

$3$ by

$1$ .

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

Step 2.9.2.2.1.2

Multiply

$- 2$ by

$2$ .

$a_{33} = 3 - 4$

Step 2.9.2.2.2

Subtract

$4$ from

$3$ .

$a_{33} = - 1$

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.

$[\begin{array}{c} - 3 & 2 & 0 \\ 8 & - 7 & 2 \\ - 3 & 3 & - 1 \end{array}]$