Algebra Examples

[\begin{array}{c} 0 & 3 & 2 \\ 4 & 3 & 3 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 0 & 3 & 2 \\ 4 & 3 & 3 \\ 1 & 2 & 0 \end{array}]$

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} 3 & 3 \\ 2 & 0 \end{array} |

$| \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} |$

Step 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

0 | \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} |

$0 | \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} |$

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} 4 & 3 \\ 1 & 0 \end{array} |

$| \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} |$

Step 1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} |

$- 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} |$

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} 4 & 3 \\ 1 & 2 \end{array} |

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

Step 1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

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

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

Step 1.9

Add the terms together.

0 | \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} | - 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} | + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 | \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} | - 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} | + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

0 | \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} | - 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} | + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 | \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} | - 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} | + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

Step 2

Multiply

0

$0$ by

| \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} |

$| \begin{array}{c} 3 & 3 \\ 2 & 0 \end{array} |$ .

0 - 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} | + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 | \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} | + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

Step 3

Evaluate

| \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} |

$| \begin{array}{c} 4 & 3 \\ 1 & 0 \end{array} |$ .

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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

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

0 - 3 (4 \cdot 0 - 1 \cdot 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 (4 \cdot 0 - 1 \cdot 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

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

4

$4$ by

0

$0$ .

0 - 3 (0 - 1 \cdot 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 (0 - 1 \cdot 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

Step 3.2.1.2

Multiply

- 1

$- 1$ by

3

$3$ .

0 - 3 (0 - 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 (0 - 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

0 - 3 (0 - 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 (0 - 3) + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

Step 3.2.2

Subtract

3

$3$ from

0

$0$ .

0 - 3 \cdot - 3 + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 \cdot - 3 + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

0 - 3 \cdot - 3 + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 \cdot - 3 + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

0 - 3 \cdot - 3 + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |

$0 - 3 \cdot - 3 + 2 | \begin{array}{c} 4 & 3 \\ 1 & 2 \end{array} |$

Step 4

Evaluate

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

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

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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

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

0 - 3 \cdot - 3 + 2 (4 \cdot 2 - 1 \cdot 3)

$0 - 3 \cdot - 3 + 2 (4 \cdot 2 - 1 \cdot 3)$

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

4

$4$ by

2

$2$ .

0 - 3 \cdot - 3 + 2 (8 - 1 \cdot 3)

$0 - 3 \cdot - 3 + 2 (8 - 1 \cdot 3)$

Step 4.2.1.2

Multiply

- 1

$- 1$ by

3

$3$ .

0 - 3 \cdot - 3 + 2 (8 - 3)

$0 - 3 \cdot - 3 + 2 (8 - 3)$

0 - 3 \cdot - 3 + 2 (8 - 3)

$0 - 3 \cdot - 3 + 2 (8 - 3)$

Step 4.2.2

Subtract

3

$3$ from

8

$8$ .

0 - 3 \cdot - 3 + 2 \cdot 5

$0 - 3 \cdot - 3 + 2 \cdot 5$

0 - 3 \cdot - 3 + 2 \cdot 5

$0 - 3 \cdot - 3 + 2 \cdot 5$

0 - 3 \cdot - 3 + 2 \cdot 5

$0 - 3 \cdot - 3 + 2 \cdot 5$

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

$- 3$ .

0 + 9 + 2 \cdot 5

$0 + 9 + 2 \cdot 5$

Step 5.1.2

Multiply

2

$2$ by

5

$5$ .

0 + 9 + 10

$0 + 9 + 10$

0 + 9 + 10

$0 + 9 + 10$

Step 5.2

Add

0

$0$ and

9

$9$ .

9 + 10

$9 + 10$

Step 5.3

Add

9

$9$ and

10

$10$ .

19

$19$

19

$19$

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