Finite Math Examples

$[\begin{array}{c} 1 & 14 & 3 & 0 \\ 0 & 20 & 2 & 3 \\ 3 & 14 & 1 & 2 \\ 2 & 12 & 0 & 1 \end{array}]$

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.

$| \begin{array}{c} 20 & 2 & 3 \\ 14 & 1 & 2 \\ 12 & 0 & 1 \end{array} |$

Step 1.4

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

$1 | \begin{array}{c} 20 & 2 & 3 \\ 14 & 1 & 2 \\ 12 & 0 & 1 \end{array} |$

Step 1.5

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

$| \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} |$

Step 1.6

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

$- 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} |$

Step 1.7

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

$| \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} |$

Step 1.8

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

$3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} |$

Step 1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 0 & 20 & 2 \\ 3 & 14 & 1 \\ 2 & 12 & 0 \end{array} |$

Step 1.10

Multiply element $a_{14}$ by its cofactor.

$0 | \begin{array}{c} 0 & 20 & 2 \\ 3 & 14 & 1 \\ 2 & 12 & 0 \end{array} |$

Step 1.11

Add the terms together.

$1 | \begin{array}{c} 20 & 2 & 3 \\ 14 & 1 & 2 \\ 12 & 0 & 1 \end{array} | - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 20 & 2 \\ 3 & 14 & 1 \\ 2 & 12 & 0 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 0 & 20 & 2 \\ 3 & 14 & 1 \\ 2 & 12 & 0 \end{array} |$ .

Step 3

Evaluate $| \begin{array}{c} 20 & 2 & 3 \\ 14 & 1 & 2 \\ 12 & 0 & 1 \end{array} |$ .

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Step 3.1

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 column $2$ by its cofactor and add.

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Step 3.1.1

Consider the corresponding sign chart.

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

Step 3.1.2

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

Step 3.1.3

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

$| \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} |$

Step 3.1.4

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

$- 2 | \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} |$

Step 3.1.5

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

$| \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} |$

Step 3.1.6

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

$1 | \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} |$

Step 3.1.7

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

$| \begin{array}{c} 20 & 3 \\ 14 & 2 \end{array} |$

Step 3.1.8

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

$0 | \begin{array}{c} 20 & 3 \\ 14 & 2 \end{array} |$

Step 3.1.9

Add the terms together.

$1 (- 2 | \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} | + 1 | \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} | + 0 | \begin{array}{c} 20 & 3 \\ 14 & 2 \end{array} |) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.2

Multiply $0$ by $| \begin{array}{c} 20 & 3 \\ 14 & 2 \end{array} |$ .

$1 (- 2 | \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} | + 1 | \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} | + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.3

Evaluate $| \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} |$ .

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Step 3.3.1

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

$1 (- 2 (14 \cdot 1 - 12 \cdot 2) + 1 | \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} | + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.3.2

Simplify the determinant.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $14$ by $1$ .

$1 (- 2 (14 - 12 \cdot 2) + 1 | \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} | + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.3.2.1.2

Multiply $- 12$ by $2$ .

$1 (- 2 (14 - 24) + 1 | \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} | + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.3.2.2

Subtract $24$ from $14$ .

$1 (- 2 \cdot - 10 + 1 | \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} | + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.4

Evaluate $| \begin{array}{c} 20 & 3 \\ 12 & 1 \end{array} |$ .

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Step 3.4.1

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

$1 (- 2 \cdot - 10 + 1 (20 \cdot 1 - 12 \cdot 3) + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.4.2

Simplify the determinant.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply $20$ by $1$ .

$1 (- 2 \cdot - 10 + 1 (20 - 12 \cdot 3) + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.4.2.1.2

Multiply $- 12$ by $3$ .

$1 (- 2 \cdot - 10 + 1 (20 - 36) + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.4.2.2

Subtract $36$ from $20$ .

$1 (- 2 \cdot - 10 + 1 \cdot - 16 + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.5

Simplify the determinant.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $- 2$ by $- 10$ .

$1 (20 + 1 \cdot - 16 + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.5.1.2

Multiply $- 16$ by $1$ .

$1 (20 - 16 + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.5.2

Subtract $16$ from $20$ .

$1 (4 + 0) - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 3.5.3

Add $4$ and $0$ .

$1 \cdot 4 - 14 | \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} | + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4

Evaluate $| \begin{array}{c} 0 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \end{array} |$ .

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Step 4.1

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 4.1.1

Consider the corresponding sign chart.

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

Step 4.1.2

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

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

$| \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$

Step 4.1.6

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

$- 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$

Step 4.1.7

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

$| \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |$

Step 4.1.8

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

$3 | \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |$

Step 4.1.9

Add the terms together.

$1 \cdot 4 - 14 (0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | - 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 3 | \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.2

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

$1 \cdot 4 - 14 (0 - 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 3 | \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.3

Evaluate $| \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$ .

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Step 4.3.1

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

$1 \cdot 4 - 14 (0 - 2 (3 \cdot 1 - 2 \cdot 2) + 3 | \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.3.2

Simplify the determinant.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $3$ by $1$ .

$1 \cdot 4 - 14 (0 - 2 (3 - 2 \cdot 2) + 3 | \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.3.2.1.2

Multiply $- 2$ by $2$ .

$1 \cdot 4 - 14 (0 - 2 (3 - 4) + 3 | \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.3.2.2

Subtract $4$ from $3$ .

$1 \cdot 4 - 14 (0 - 2 \cdot - 1 + 3 | \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.4

Evaluate $| \begin{array}{c} 3 & 1 \\ 2 & 0 \end{array} |$ .

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Step 4.4.1

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

$1 \cdot 4 - 14 (0 - 2 \cdot - 1 + 3 (3 \cdot 0 - 2 \cdot 1)) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.4.2

Simplify the determinant.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Multiply $3$ by $0$ .

$1 \cdot 4 - 14 (0 - 2 \cdot - 1 + 3 (0 - 2 \cdot 1)) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.4.2.1.2

Multiply $- 2$ by $1$ .

$1 \cdot 4 - 14 (0 - 2 \cdot - 1 + 3 (0 - 2)) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.4.2.2

Subtract $2$ from $0$ .

$1 \cdot 4 - 14 (0 - 2 \cdot - 1 + 3 \cdot - 2) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.5

Simplify the determinant.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $- 2$ by $- 1$ .

$1 \cdot 4 - 14 (0 + 2 + 3 \cdot - 2) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.5.1.2

Multiply $3$ by $- 2$ .

$1 \cdot 4 - 14 (0 + 2 - 6) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.5.2

Add $0$ and $2$ .

$1 \cdot 4 - 14 (2 - 6) + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 4.5.3

Subtract $6$ from $2$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 | \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} | + 0$

Step 5

Evaluate $| \begin{array}{c} 0 & 20 & 3 \\ 3 & 14 & 2 \\ 2 & 12 & 1 \end{array} |$ .

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Step 5.1

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 5.1.1

Consider the corresponding sign chart.

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

Step 5.1.2

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

Step 5.1.3

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

$| \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} |$

Step 5.1.4

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

$0 | \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} |$

Step 5.1.5

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

$| \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$

Step 5.1.6

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

$- 20 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$

Step 5.1.7

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

$| \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |$

Step 5.1.8

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

$3 | \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |$

Step 5.1.9

Add the terms together.

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 | \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} | - 20 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 3 | \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |) + 0$

Step 5.2

Multiply $0$ by $| \begin{array}{c} 14 & 2 \\ 12 & 1 \end{array} |$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 3 | \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |) + 0$

Step 5.3

Evaluate $| \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$ .

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Step 5.3.1

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

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 (3 \cdot 1 - 2 \cdot 2) + 3 | \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |) + 0$

Step 5.3.2

Simplify the determinant.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply $3$ by $1$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 (3 - 2 \cdot 2) + 3 | \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |) + 0$

Step 5.3.2.1.2

Multiply $- 2$ by $2$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 (3 - 4) + 3 | \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |) + 0$

Step 5.3.2.2

Subtract $4$ from $3$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 \cdot - 1 + 3 | \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |) + 0$

Step 5.4

Evaluate $| \begin{array}{c} 3 & 14 \\ 2 & 12 \end{array} |$ .

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Step 5.4.1

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

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 \cdot - 1 + 3 (3 \cdot 12 - 2 \cdot 14)) + 0$

Step 5.4.2

Simplify the determinant.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Multiply $3$ by $12$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 \cdot - 1 + 3 (36 - 2 \cdot 14)) + 0$

Step 5.4.2.1.2

Multiply $- 2$ by $14$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 \cdot - 1 + 3 (36 - 28)) + 0$

Step 5.4.2.2

Subtract $28$ from $36$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 - 20 \cdot - 1 + 3 \cdot 8) + 0$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply $- 20$ by $- 1$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 + 20 + 3 \cdot 8) + 0$

Step 5.5.1.2

Multiply $3$ by $8$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (0 + 20 + 24) + 0$

Step 5.5.2

Add $0$ and $20$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 (20 + 24) + 0$

Step 5.5.3

Add $20$ and $24$ .

$1 \cdot 4 - 14 \cdot - 4 + 3 \cdot 44 + 0$

Step 6

Simplify the determinant.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $4$ by $1$ .

$4 - 14 \cdot - 4 + 3 \cdot 44 + 0$

Step 6.1.2

Multiply $- 14$ by $- 4$ .

$4 + 56 + 3 \cdot 44 + 0$

Step 6.1.3

Multiply $3$ by $44$ .

$4 + 56 + 132 + 0$

Step 6.2

Add $4$ and $56$ .

$60 + 132 + 0$

Step 6.3

Add $60$ and $132$ .

$192 + 0$

Step 6.4

Add $192$ and $0$ .

$192$