Linear Algebra Examples

$[\begin{array}{c} 2 & 3 & 11 & 5 \\ 1 & 1 & 5 & 2 \\ 2 & 1 & 3 & 2 \\ 1 & 1 & 3 & 4 \end{array}]$

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.

$| \begin{array}{c} 1 & 5 & 2 \\ 1 & 3 & 2 \\ 1 & 3 & 4 \end{array} |$

Step 1.4

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

$2 | \begin{array}{c} 1 & 5 & 2 \\ 1 & 3 & 2 \\ 1 & 3 & 4 \end{array} |$

Step 1.5

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

$| \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} |$

Step 1.6

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

$- 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} |$

Step 1.7

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

$| \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} |$

Step 1.8

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

$11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} |$

Step 1.9

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

$| \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 1.10

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

$- 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 1.11

Add the terms together.

$2 | \begin{array}{c} 1 & 5 & 2 \\ 1 & 3 & 2 \\ 1 & 3 & 4 \end{array} | - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 1 & 5 & 2 \\ 1 & 3 & 2 \\ 1 & 3 & 4 \end{array} |$ .

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Step 2.1

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 2.1.1

Consider the corresponding sign chart.

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

Step 2.1.2

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

Step 2.1.3

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

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

Step 2.1.4

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

$1 | \begin{array}{c} 3 & 2 \\ 3 & 4 \end{array} |$

Step 2.1.5

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

$| \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} |$

Step 2.1.6

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

$- 5 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} |$

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.1.9

Add the terms together.

$2 (1 | \begin{array}{c} 3 & 2 \\ 3 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.2

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

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Step 2.2.1

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

$2 (1 (3 \cdot 4 - 3 \cdot 2) - 5 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.2.2

Simplify the determinant.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $3$ by $4$ .

$2 (1 (12 - 3 \cdot 2) - 5 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.2.2.1.2

Multiply $- 3$ by $2$ .

$2 (1 (12 - 6) - 5 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.2.2.2

Subtract $6$ from $12$ .

$2 (1 \cdot 6 - 5 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.3

Evaluate $| \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} |$ .

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Step 2.3.1

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

$2 (1 \cdot 6 - 5 (1 \cdot 4 - 1 \cdot 2) + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.3.2

Simplify the determinant.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $4$ by $1$ .

$2 (1 \cdot 6 - 5 (4 - 1 \cdot 2) + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.3.2.1.2

Multiply $- 1$ by $2$ .

$2 (1 \cdot 6 - 5 (4 - 2) + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.3.2.2

Subtract $2$ from $4$ .

$2 (1 \cdot 6 - 5 \cdot 2 + 2 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} |) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.4

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

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Step 2.4.1

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

$2 (1 \cdot 6 - 5 \cdot 2 + 2 (1 \cdot 3 - 1 \cdot 3)) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.4.2

Simplify the determinant.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $3$ by $1$ .

$2 (1 \cdot 6 - 5 \cdot 2 + 2 (3 - 1 \cdot 3)) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.4.2.1.2

Multiply $- 1$ by $3$ .

$2 (1 \cdot 6 - 5 \cdot 2 + 2 (3 - 3)) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.4.2.2

Subtract $3$ from $3$ .

$2 (1 \cdot 6 - 5 \cdot 2 + 2 \cdot 0) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.5

Simplify the determinant.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $6$ by $1$ .

$2 (6 - 5 \cdot 2 + 2 \cdot 0) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.5.1.2

Multiply $- 5$ by $2$ .

$2 (6 - 10 + 2 \cdot 0) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.5.1.3

Multiply $2$ by $0$ .

$2 (6 - 10 + 0) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.5.2

Subtract $10$ from $6$ .

$2 (- 4 + 0) - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 2.5.3

Add $- 4$ and $0$ .

$2 \cdot - 4 - 3 | \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} | + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1 & 5 & 2 \\ 2 & 3 & 2 \\ 1 & 3 & 4 \end{array} |$ .

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

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

Step 3.1.4

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

$1 | \begin{array}{c} 3 & 2 \\ 3 & 4 \end{array} |$

Step 3.1.5

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

$| \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} |$

Step 3.1.6

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

$- 5 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} |$

Step 3.1.7

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

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

Step 3.1.8

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

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

Step 3.1.9

Add the terms together.

$2 \cdot - 4 - 3 (1 | \begin{array}{c} 3 & 2 \\ 3 & 4 \end{array} | - 5 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.2

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

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Step 3.2.1

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

$2 \cdot - 4 - 3 (1 (3 \cdot 4 - 3 \cdot 2) - 5 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.2.2

Simplify the determinant.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $3$ by $4$ .

$2 \cdot - 4 - 3 (1 (12 - 3 \cdot 2) - 5 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.2.2.1.2

Multiply $- 3$ by $2$ .

$2 \cdot - 4 - 3 (1 (12 - 6) - 5 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.2.2.2

Subtract $6$ from $12$ .

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.3

Evaluate $| \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} |$ .

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Step 3.3.1

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

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 (2 \cdot 4 - 1 \cdot 2) + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

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 $2$ by $4$ .

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 (8 - 1 \cdot 2) + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.3.2.1.2

Multiply $- 1$ by $2$ .

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 (8 - 2) + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.3.2.2

Subtract $2$ from $8$ .

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 \cdot 6 + 2 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} |) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.4

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

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Step 3.4.1

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

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 \cdot 6 + 2 (2 \cdot 3 - 1 \cdot 3)) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

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 $2$ by $3$ .

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 \cdot 6 + 2 (6 - 1 \cdot 3)) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.4.2.1.2

Multiply $- 1$ by $3$ .

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 \cdot 6 + 2 (6 - 3)) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.4.2.2

Subtract $3$ from $6$ .

$2 \cdot - 4 - 3 (1 \cdot 6 - 5 \cdot 6 + 2 \cdot 3) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

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 $6$ by $1$ .

$2 \cdot - 4 - 3 (6 - 5 \cdot 6 + 2 \cdot 3) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.5.1.2

Multiply $- 5$ by $6$ .

$2 \cdot - 4 - 3 (6 - 30 + 2 \cdot 3) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.5.1.3

Multiply $2$ by $3$ .

$2 \cdot - 4 - 3 (6 - 30 + 6) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.5.2

Subtract $30$ from $6$ .

$2 \cdot - 4 - 3 (- 24 + 6) + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 3.5.3

Add $- 24$ and $6$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 | \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} | - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 1 & 1 & 4 \end{array} |$ .

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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.

$| \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} |$

Step 4.1.4

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

$1 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} |$

Step 4.1.5

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

$| \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} |$

Step 4.1.6

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

$- 1 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} |$

Step 4.1.7

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

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

Step 4.1.8

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

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

Step 4.1.9

Add the terms together.

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 | \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} | - 1 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.2

Evaluate $| \begin{array}{c} 1 & 2 \\ 1 & 4 \end{array} |$ .

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Step 4.2.1

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

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 (1 \cdot 4 - 1 \cdot 2) - 1 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.2.2

Simplify the determinant.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $4$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 (4 - 1 \cdot 2) - 1 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.2.2.1.2

Multiply $- 1$ by $2$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 (4 - 2) - 1 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.2.2.2

Subtract $2$ from $4$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 | \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} | + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.3

Evaluate $| \begin{array}{c} 2 & 2 \\ 1 & 4 \end{array} |$ .

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Step 4.3.1

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

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 (2 \cdot 4 - 1 \cdot 2) + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

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 $2$ by $4$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 (8 - 1 \cdot 2) + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.3.2.1.2

Multiply $- 1$ by $2$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 (8 - 2) + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.3.2.2

Subtract $2$ from $8$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 \cdot 6 + 2 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.4

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

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Step 4.4.1

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

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 \cdot 6 + 2 (2 \cdot 1 - 1 \cdot 1)) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

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 $2$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 \cdot 6 + 2 (2 - 1 \cdot 1)) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.4.2.1.2

Multiply $- 1$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 \cdot 6 + 2 (2 - 1)) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.4.2.2

Subtract $1$ from $2$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (1 \cdot 2 - 1 \cdot 6 + 2 \cdot 1) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

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 $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (2 - 1 \cdot 6 + 2 \cdot 1) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.5.1.2

Multiply $- 1$ by $6$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (2 - 6 + 2 \cdot 1) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.5.1.3

Multiply $2$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (2 - 6 + 2) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.5.2

Subtract $6$ from $2$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 (- 4 + 2) - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 4.5.3

Add $- 4$ and $2$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 | \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 1 & 1 & 5 \\ 2 & 1 & 3 \\ 1 & 1 & 3 \end{array} |$ .

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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.

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.1.7

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

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

Step 5.1.8

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

$5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |$

Step 5.1.9

Add the terms together.

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 | \begin{array}{c} 1 & 3 \\ 1 & 3 \end{array} | - 1 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} | + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.2

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

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Step 5.2.1

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

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 (1 \cdot 3 - 1 \cdot 3) - 1 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} | + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.2.2

Simplify the determinant.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $3$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 (3 - 1 \cdot 3) - 1 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} | + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.2.2.1.2

Multiply $- 1$ by $3$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 (3 - 3) - 1 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} | + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.2.2.2

Subtract $3$ from $3$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 | \begin{array}{c} 2 & 3 \\ 1 & 3 \end{array} | + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.3

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

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Step 5.3.1

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

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 (2 \cdot 3 - 1 \cdot 3) + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

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 $2$ by $3$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 (6 - 1 \cdot 3) + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.3.2.1.2

Multiply $- 1$ by $3$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 (6 - 3) + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.3.2.2

Subtract $3$ from $6$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 \cdot 3 + 5 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |)$

Step 5.4

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

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Step 5.4.1

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

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 \cdot 3 + 5 (2 \cdot 1 - 1 \cdot 1))$

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 $2$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 \cdot 3 + 5 (2 - 1 \cdot 1))$

Step 5.4.2.1.2

Multiply $- 1$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 \cdot 3 + 5 (2 - 1))$

Step 5.4.2.2

Subtract $1$ from $2$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (1 \cdot 0 - 1 \cdot 3 + 5 \cdot 1)$

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 $0$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (0 - 1 \cdot 3 + 5 \cdot 1)$

Step 5.5.1.2

Multiply $- 1$ by $3$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (0 - 3 + 5 \cdot 1)$

Step 5.5.1.3

Multiply $5$ by $1$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (0 - 3 + 5)$

Step 5.5.2

Subtract $3$ from $0$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 (- 3 + 5)$

Step 5.5.3

Add $- 3$ and $5$ .

$2 \cdot - 4 - 3 \cdot - 18 + 11 \cdot - 2 - 5 \cdot 2$

Step 6

Simplify the determinant.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $2$ by $- 4$ .

$- 8 - 3 \cdot - 18 + 11 \cdot - 2 - 5 \cdot 2$

Step 6.1.2

Multiply $- 3$ by $- 18$ .

$- 8 + 54 + 11 \cdot - 2 - 5 \cdot 2$

Step 6.1.3

Multiply $11$ by $- 2$ .

$- 8 + 54 - 22 - 5 \cdot 2$

Step 6.1.4

Multiply $- 5$ by $2$ .

$- 8 + 54 - 22 - 10$

Step 6.2

Add $- 8$ and $54$ .

$46 - 22 - 10$

Step 6.3

Subtract $22$ from $46$ .

$24 - 10$

Step 6.4

Subtract $10$ from $24$ .

$14$