Precalculus Examples

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

Step 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $3$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

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

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 $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

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

Step 1.4

Multiply element $a_{31}$ by its cofactor.

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

Step 1.5

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

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

Step 1.6

Multiply element $a_{32}$ by its cofactor.

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

Step 1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

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

Step 1.8

Multiply element $a_{33}$ by its cofactor.

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

Step 1.9

Add the terms together.

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

Step 2

Multiply $0$ by $| \begin{array}{c} - \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{array} |$ .

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

Step 3

Multiply $0$ by $| \begin{array}{c} \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{2} & 0 \end{array} |$ .

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

Step 4

Evaluate $| \begin{array}{c} \frac{1}{2} & 0 \\ - \frac{1}{2} & \frac{1}{2} \end{array} |$ .

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Step 4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 - \frac{1}{2} (\frac{1}{2} \cdot \frac{1}{2} - (- \frac{1}{2} \cdot 0)) + 0$

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 $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 4.2.1.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$0 - \frac{1}{2} (\frac{1}{2 \cdot 2} - (- \frac{1}{2} \cdot 0)) + 0$

Step 4.2.1.1.2

Multiply $2$ by $2$ .

$0 - \frac{1}{2} (\frac{1}{4} - (- \frac{1}{2} \cdot 0)) + 0$

Step 4.2.1.2

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 4.2.1.2.1

Multiply $0$ by $- 1$ .

$0 - \frac{1}{2} (\frac{1}{4} - (0 (\frac{1}{2}))) + 0$

Step 4.2.1.2.2

Multiply $0$ by $\frac{1}{2}$ .

$0 - \frac{1}{2} (\frac{1}{4} - 0) + 0$

Step 4.2.1.3

Multiply $- 1$ by $0$ .

$0 - \frac{1}{2} (\frac{1}{4} + 0) + 0$

Step 4.2.2

Add $\frac{1}{4}$ and $0$ .

$0 - \frac{1}{2} \cdot \frac{1}{4} + 0$

Step 5

Simplify the determinant.

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Step 5.1

Multiply $- \frac{1}{2} \cdot \frac{1}{4}$ .

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Step 5.1.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$0 - \frac{1}{4 \cdot 2} + 0$

Step 5.1.2

Multiply $4$ by $2$ .

$0 - \frac{1}{8} + 0$

Step 5.2

Subtract $\frac{1}{8}$ from $0$ .

$- \frac{1}{8} + 0$

Step 5.3

Add $- \frac{1}{8}$ and $0$ .

$- \frac{1}{8}$