Linear Algebra Examples

det

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

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

Step 1

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in row

1

$1$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

Step 1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

| \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |

$| \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Step 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |

$3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Step 1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

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

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

Step 1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

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

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

Step 1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

| \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$| \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 1.9

Add the terms together.

3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} | - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} | - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} | - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} | - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 2

Evaluate

| \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |

$| \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$ .

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Step 2.1

The determinant of a

2 \times 2

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

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

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

3 (4 \cdot 8 - 7 \cdot 5) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 (4 \cdot 8 - 7 \cdot 5) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply

4

$4$ by

8

$8$ .

3 (32 - 7 \cdot 5) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 (32 - 7 \cdot 5) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 2.2.1.2

Multiply

- 7

$- 7$ by

5

$5$ .

3 (32 - 35) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 (32 - 35) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

3 (32 - 35) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 (32 - 35) - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 2.2.2

Subtract

35

$35$ from

32

$32$ .

3 \cdot - 3 - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

3 \cdot - 3 - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

3 \cdot - 3 - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 | \begin{array}{c} 3 & 5 \\ 3 & 8 \end{array} | + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 3

Evaluate

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

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

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Step 3.1

The determinant of a

2 \times 2

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

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

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

3 \cdot - 3 - 2 (3 \cdot 8 - 3 \cdot 5) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 (3 \cdot 8 - 3 \cdot 5) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 3.2

Simplify the determinant.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply

3

$3$ by

8

$8$ .

3 \cdot - 3 - 2 (24 - 3 \cdot 5) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 (24 - 3 \cdot 5) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 3.2.1.2

Multiply

- 3

$- 3$ by

5

$5$ .

3 \cdot - 3 - 2 (24 - 15) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 (24 - 15) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

3 \cdot - 3 - 2 (24 - 15) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 (24 - 15) + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 3.2.2

Subtract

15

$15$ from

24

$24$ .

3 \cdot - 3 - 2 \cdot 9 + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 \cdot 9 + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

3 \cdot - 3 - 2 \cdot 9 + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 \cdot 9 + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

3 \cdot - 3 - 2 \cdot 9 + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$3 \cdot - 3 - 2 \cdot 9 + 1 | \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$

Step 4

Evaluate

| \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |

$| \begin{array}{c} 3 & 4 \\ 3 & 7 \end{array} |$ .

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Step 4.1

The determinant of a

2 \times 2

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

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

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

3 \cdot - 3 - 2 \cdot 9 + 1 (3 \cdot 7 - 3 \cdot 4)

$3 \cdot - 3 - 2 \cdot 9 + 1 (3 \cdot 7 - 3 \cdot 4)$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply

3

$3$ by

7

$7$ .

3 \cdot - 3 - 2 \cdot 9 + 1 (21 - 3 \cdot 4)

$3 \cdot - 3 - 2 \cdot 9 + 1 (21 - 3 \cdot 4)$

Step 4.2.1.2

Multiply

- 3

$- 3$ by

4

$4$ .

3 \cdot - 3 - 2 \cdot 9 + 1 (21 - 12)

$3 \cdot - 3 - 2 \cdot 9 + 1 (21 - 12)$

3 \cdot - 3 - 2 \cdot 9 + 1 (21 - 12)

$3 \cdot - 3 - 2 \cdot 9 + 1 (21 - 12)$

Step 4.2.2

Subtract

12

$12$ from

21

$21$ .

3 \cdot - 3 - 2 \cdot 9 + 1 \cdot 9

$3 \cdot - 3 - 2 \cdot 9 + 1 \cdot 9$

3 \cdot - 3 - 2 \cdot 9 + 1 \cdot 9

$3 \cdot - 3 - 2 \cdot 9 + 1 \cdot 9$

3 \cdot - 3 - 2 \cdot 9 + 1 \cdot 9

$3 \cdot - 3 - 2 \cdot 9 + 1 \cdot 9$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply

3

$3$ by

- 3

$- 3$ .

- 9 - 2 \cdot 9 + 1 \cdot 9

$- 9 - 2 \cdot 9 + 1 \cdot 9$

Step 5.1.2

Multiply

- 2

$- 2$ by

9

$9$ .

- 9 - 18 + 1 \cdot 9

$- 9 - 18 + 1 \cdot 9$

Step 5.1.3

Multiply

9

$9$ by

1

$1$ .

- 9 - 18 + 9

$- 9 - 18 + 9$

- 9 - 18 + 9

$- 9 - 18 + 9$

Step 5.2

Subtract

18

$18$ from

- 9

$- 9$ .

- 27 + 9

$- 27 + 9$

Step 5.3

Add

- 27

$- 27$ and

9

$9$ .

- 18

$- 18$

- 18

$- 18$