Finite Math Examples

$5 [\begin{array}{c} 8 & 3 & 1 \\ 2 & 4 & 3 \\ 3 & 2 & 4 \end{array}]$

Step 1

Multiply $5$ by each element of the matrix.

$[\begin{array}{c} 5 \cdot 8 & 5 \cdot 3 & 5 \cdot 1 \\ 5 \cdot 2 & 5 \cdot 4 & 5 \cdot 3 \\ 5 \cdot 3 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2

Simplify each element in the matrix.

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Step 2.1

Multiply $5$ by $8$ .

$[\begin{array}{c} 40 & 5 \cdot 3 & 5 \cdot 1 \\ 5 \cdot 2 & 5 \cdot 4 & 5 \cdot 3 \\ 5 \cdot 3 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2.2

Multiply $5$ by $3$ .

$[\begin{array}{c} 40 & 15 & 5 \cdot 1 \\ 5 \cdot 2 & 5 \cdot 4 & 5 \cdot 3 \\ 5 \cdot 3 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2.3

Multiply $5$ by $1$ .

$[\begin{array}{c} 40 & 15 & 5 \\ 5 \cdot 2 & 5 \cdot 4 & 5 \cdot 3 \\ 5 \cdot 3 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2.4

Multiply $5$ by $2$ .

$[\begin{array}{c} 40 & 15 & 5 \\ 10 & 5 \cdot 4 & 5 \cdot 3 \\ 5 \cdot 3 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2.5

Multiply $5$ by $4$ .

$[\begin{array}{c} 40 & 15 & 5 \\ 10 & 20 & 5 \cdot 3 \\ 5 \cdot 3 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2.6

Multiply $5$ by $3$ .

$[\begin{array}{c} 40 & 15 & 5 \\ 10 & 20 & 15 \\ 5 \cdot 3 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2.7

Multiply $5$ by $3$ .

$[\begin{array}{c} 40 & 15 & 5 \\ 10 & 20 & 15 \\ 15 & 5 \cdot 2 & 5 \cdot 4 \end{array}]$

Step 2.8

Multiply $5$ by $2$ .

$[\begin{array}{c} 40 & 15 & 5 \\ 10 & 20 & 15 \\ 15 & 10 & 5 \cdot 4 \end{array}]$

Step 2.9

Multiply $5$ by $4$ .

$[\begin{array}{c} 40 & 15 & 5 \\ 10 & 20 & 15 \\ 15 & 10 & 20 \end{array}]$

Step 3

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 3.1

Consider the corresponding sign chart.

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

Step 3.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 3.3

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

$| \begin{array}{c} 20 & 15 \\ 10 & 20 \end{array} |$

Step 3.4

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

$40 | \begin{array}{c} 20 & 15 \\ 10 & 20 \end{array} |$

Step 3.5

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

$| \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} |$

Step 3.6

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

$- 15 | \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} |$

Step 3.7

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

$| \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 3.8

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

$5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 3.9

Add the terms together.

$40 | \begin{array}{c} 20 & 15 \\ 10 & 20 \end{array} | - 15 | \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} | + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 20 & 15 \\ 10 & 20 \end{array} |$ .

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Step 4.1

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

$40 (20 \cdot 20 - 10 \cdot 15) - 15 | \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} | + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

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 $20$ by $20$ .

$40 (400 - 10 \cdot 15) - 15 | \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} | + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 4.2.1.2

Multiply $- 10$ by $15$ .

$40 (400 - 150) - 15 | \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} | + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 4.2.2

Subtract $150$ from $400$ .

$40 \cdot 250 - 15 | \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} | + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 10 & 15 \\ 15 & 20 \end{array} |$ .

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Step 5.1

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

$40 \cdot 250 - 15 (10 \cdot 20 - 15 \cdot 15) + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 5.2

Simplify the determinant.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $10$ by $20$ .

$40 \cdot 250 - 15 (200 - 15 \cdot 15) + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 5.2.1.2

Multiply $- 15$ by $15$ .

$40 \cdot 250 - 15 (200 - 225) + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 5.2.2

Subtract $225$ from $200$ .

$40 \cdot 250 - 15 \cdot - 25 + 5 | \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$

Step 6

Evaluate $| \begin{array}{c} 10 & 20 \\ 15 & 10 \end{array} |$ .

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Step 6.1

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

$40 \cdot 250 - 15 \cdot - 25 + 5 (10 \cdot 10 - 15 \cdot 20)$

Step 6.2

Simplify the determinant.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $10$ by $10$ .

$40 \cdot 250 - 15 \cdot - 25 + 5 (100 - 15 \cdot 20)$

Step 6.2.1.2

Multiply $- 15$ by $20$ .

$40 \cdot 250 - 15 \cdot - 25 + 5 (100 - 300)$

Step 6.2.2

Subtract $300$ from $100$ .

$40 \cdot 250 - 15 \cdot - 25 + 5 \cdot - 200$

Step 7

Simplify the determinant.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $40$ by $250$ .

$10000 - 15 \cdot - 25 + 5 \cdot - 200$

Step 7.1.2

Multiply $- 15$ by $- 25$ .

$10000 + 375 + 5 \cdot - 200$

Step 7.1.3

Multiply $5$ by $- 200$ .

$10000 + 375 - 1000$

Step 7.2

Add $10000$ and $375$ .

$10375 - 1000$

Step 7.3

Subtract $1000$ from $10375$ .

$9375$