Examples

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

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

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 column

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

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

Step 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

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

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

Step 1.5

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

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

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

Step 1.6

Multiply element

a_{21}

$a_{21}$ by its cofactor.

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

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

Step 1.7

The minor for

a_{31}

$a_{31}$ is the determinant with row

3

$3$ and column

1

$1$ deleted.

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

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

Step 1.8

Multiply element

a_{31}

$a_{31}$ by its cofactor.

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

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

Step 1.9

Add the terms together.

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

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

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

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

Step 2

Multiply

0

$0$ by

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

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

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

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

Step 3

Multiply

0

$0$ by

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

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

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

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

Step 4

Evaluate

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

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

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

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

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

1

$1$ .

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

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

Step 4.2.1.2

Multiply

0

$0$ by

3

$3$ .

0 - 2 (3 + 0) + 0

$0 - 2 (3 + 0) + 0$

0 - 2 (3 + 0) + 0

$0 - 2 (3 + 0) + 0$

Step 4.2.2

Add

3

$3$ and

0

$0$ .

0 - 2 \cdot 3 + 0

$0 - 2 \cdot 3 + 0$

0 - 2 \cdot 3 + 0

$0 - 2 \cdot 3 + 0$

0 - 2 \cdot 3 + 0

$0 - 2 \cdot 3 + 0$

Step 5

Simplify the determinant.

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Step 5.1

Multiply

- 2

$- 2$ by

3

$3$ .

0 - 6 + 0

$0 - 6 + 0$

Step 5.2

Subtract

6

$6$ from

0

$0$ .

- 6 + 0

$- 6 + 0$

Step 5.3

Add

- 6

$- 6$ and

0

$0$ .

- 6

$- 6$

- 6

$- 6$

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