Precalculus Examples

$[\begin{array}{c} 6 & 9 & 2 \\ 9 & 9 & 8 \\ 1 & 2 & 10 \end{array}]$

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 $1$ 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 $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 9 & 8 \\ 2 & 10 \end{array} |$

Step 1.4

Multiply element $a_{11}$ by its cofactor.

$6 | \begin{array}{c} 9 & 8 \\ 2 & 10 \end{array} |$

Step 1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} |$

Step 1.6

Multiply element $a_{12}$ by its cofactor.

$- 9 | \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} |$

Step 1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 1.8

Multiply element $a_{13}$ by its cofactor.

$2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 1.9

Add the terms together.

$6 | \begin{array}{c} 9 & 8 \\ 2 & 10 \end{array} | - 9 | \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} | + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 9 & 8 \\ 2 & 10 \end{array} |$ .

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Step 2.1

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

$6 (9 \cdot 10 - 2 \cdot 8) - 9 | \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} | + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

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 $9$ by $10$ .

$6 (90 - 2 \cdot 8) - 9 | \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} | + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 2.2.1.2

Multiply $- 2$ by $8$ .

$6 (90 - 16) - 9 | \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} | + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 2.2.2

Subtract $16$ from $90$ .

$6 \cdot 74 - 9 | \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} | + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 9 & 8 \\ 1 & 10 \end{array} |$ .

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Step 3.1

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

$6 \cdot 74 - 9 (9 \cdot 10 - 1 \cdot 8) + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

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 $9$ by $10$ .

$6 \cdot 74 - 9 (90 - 1 \cdot 8) + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 3.2.1.2

Multiply $- 1$ by $8$ .

$6 \cdot 74 - 9 (90 - 8) + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 3.2.2

Subtract $8$ from $90$ .

$6 \cdot 74 - 9 \cdot 82 + 2 | \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 9 & 9 \\ 1 & 2 \end{array} |$ .

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Step 4.1

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

$6 \cdot 74 - 9 \cdot 82 + 2 (9 \cdot 2 - 1 \cdot 9)$

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 $9$ by $2$ .

$6 \cdot 74 - 9 \cdot 82 + 2 (18 - 1 \cdot 9)$

Step 4.2.1.2

Multiply $- 1$ by $9$ .

$6 \cdot 74 - 9 \cdot 82 + 2 (18 - 9)$

Step 4.2.2

Subtract $9$ from $18$ .

$6 \cdot 74 - 9 \cdot 82 + 2 \cdot 9$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $6$ by $74$ .

$444 - 9 \cdot 82 + 2 \cdot 9$

Step 5.1.2

Multiply $- 9$ by $82$ .

$444 - 738 + 2 \cdot 9$

Step 5.1.3

Multiply $2$ by $9$ .

$444 - 738 + 18$

Step 5.2

Subtract $738$ from $444$ .

$- 294 + 18$

Step 5.3

Add $- 294$ and $18$ .

$- 276$