Finite Math Examples

$[\begin{array}{c} 2 & 2 & - 6 \\ 8 & 9 & - 4 \\ 6 & 2 & 2 \end{array}]$

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.

$| \begin{array}{c} 9 & - 4 \\ 2 & 2 \end{array} |$

Step 1.4

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

$2 | \begin{array}{c} 9 & - 4 \\ 2 & 2 \end{array} |$

Step 1.5

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

$| \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} |$

Step 1.6

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

$- 2 | \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} |$

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Add the terms together.

$2 | \begin{array}{c} 9 & - 4 \\ 2 & 2 \end{array} | - 2 | \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} | - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 9 & - 4 \\ 2 & 2 \end{array} |$ .

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Step 2.1

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

$2 (9 \cdot 2 - 2 \cdot - 4) - 2 | \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} | - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

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 $2$ .

$2 (18 - 2 \cdot - 4) - 2 | \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} | - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

Step 2.2.1.2

Multiply $- 2$ by $- 4$ .

$2 (18 + 8) - 2 | \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} | - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

Step 2.2.2

Add $18$ and $8$ .

$2 \cdot 26 - 2 | \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} | - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 8 & - 4 \\ 6 & 2 \end{array} |$ .

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Step 3.1

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

$2 \cdot 26 - 2 (8 \cdot 2 - 6 \cdot - 4) - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

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 $8$ by $2$ .

$2 \cdot 26 - 2 (16 - 6 \cdot - 4) - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

Step 3.2.1.2

Multiply $- 6$ by $- 4$ .

$2 \cdot 26 - 2 (16 + 24) - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

Step 3.2.2

Add $16$ and $24$ .

$2 \cdot 26 - 2 \cdot 40 - 6 | \begin{array}{c} 8 & 9 \\ 6 & 2 \end{array} |$

Step 4

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

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Step 4.1

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

$2 \cdot 26 - 2 \cdot 40 - 6 (8 \cdot 2 - 6 \cdot 9)$

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 $8$ by $2$ .

$2 \cdot 26 - 2 \cdot 40 - 6 (16 - 6 \cdot 9)$

Step 4.2.1.2

Multiply $- 6$ by $9$ .

$2 \cdot 26 - 2 \cdot 40 - 6 (16 - 54)$

Step 4.2.2

Subtract $54$ from $16$ .

$2 \cdot 26 - 2 \cdot 40 - 6 \cdot - 38$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $2$ by $26$ .

$52 - 2 \cdot 40 - 6 \cdot - 38$

Step 5.1.2

Multiply $- 2$ by $40$ .

$52 - 80 - 6 \cdot - 38$

Step 5.1.3

Multiply $- 6$ by $- 38$ .

$52 - 80 + 228$

Step 5.2

Subtract $80$ from $52$ .

$- 28 + 228$

Step 5.3

Add $- 28$ and $228$ .

$200$