Finite Math Examples

$[\begin{array}{c} 2 & - 14 & - 4 \\ 2 & 6 & 0 \\ - 3 & 23 & 5 \end{array}]$

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 $2$ 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 $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} - 14 & - 4 \\ 23 & 5 \end{array} |$

Step 1.4

Multiply element $a_{21}$ by its cofactor.

$- 2 | \begin{array}{c} - 14 & - 4 \\ 23 & 5 \end{array} |$

Step 1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} |$

Step 1.6

Multiply element $a_{22}$ by its cofactor.

$6 | \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} |$

Step 1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 2 & - 14 \\ - 3 & 23 \end{array} |$

Step 1.8

Multiply element $a_{23}$ by its cofactor.

$0 | \begin{array}{c} 2 & - 14 \\ - 3 & 23 \end{array} |$

Step 1.9

Add the terms together.

$- 2 | \begin{array}{c} - 14 & - 4 \\ 23 & 5 \end{array} | + 6 | \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} | + 0 | \begin{array}{c} 2 & - 14 \\ - 3 & 23 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 2 & - 14 \\ - 3 & 23 \end{array} |$ .

$- 2 | \begin{array}{c} - 14 & - 4 \\ 23 & 5 \end{array} | + 6 | \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} | + 0$

Step 3

Evaluate $| \begin{array}{c} - 14 & - 4 \\ 23 & 5 \end{array} |$ .

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Step 3.1

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

$- 2 (- 14 \cdot 5 - 23 \cdot - 4) + 6 | \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} | + 0$

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 $- 14$ by $5$ .

$- 2 (- 70 - 23 \cdot - 4) + 6 | \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} | + 0$

Step 3.2.1.2

Multiply $- 23$ by $- 4$ .

$- 2 (- 70 + 92) + 6 | \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} | + 0$

Step 3.2.2

Add $- 70$ and $92$ .

$- 2 \cdot 22 + 6 | \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} | + 0$

Step 4

Evaluate $| \begin{array}{c} 2 & - 4 \\ - 3 & 5 \end{array} |$ .

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Step 4.1

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

$- 2 \cdot 22 + 6 (2 \cdot 5 - (- 3 \cdot - 4)) + 0$

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 $2$ by $5$ .

$- 2 \cdot 22 + 6 (10 - (- 3 \cdot - 4)) + 0$

Step 4.2.1.2

Multiply $- (- 3 \cdot - 4)$ .

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Step 4.2.1.2.1

Multiply $- 3$ by $- 4$ .

$- 2 \cdot 22 + 6 (10 - 1 \cdot 12) + 0$

Step 4.2.1.2.2

Multiply $- 1$ by $12$ .

$- 2 \cdot 22 + 6 (10 - 12) + 0$

Step 4.2.2

Subtract $12$ from $10$ .

$- 2 \cdot 22 + 6 \cdot - 2 + 0$

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 $22$ .

$- 44 + 6 \cdot - 2 + 0$

Step 5.1.2

Multiply $6$ by $- 2$ .

$- 44 - 12 + 0$

Step 5.2

Subtract $12$ from $- 44$ .

$- 56 + 0$

Step 5.3

Add $- 56$ and $0$ .

$- 56$