Linear Algebra Examples

$[\begin{array}{c} 2 & - 1 & 2 \\ 12 & 1 & - 3 \\ 6 & - 1 & 6 \end{array}]$

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.

$| \begin{array}{c} 1 & - 3 \\ - 1 & 6 \end{array} |$

Step 1.4

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

$2 | \begin{array}{c} 1 & - 3 \\ - 1 & 6 \end{array} |$

Step 1.5

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

$| \begin{array}{c} 12 & - 3 \\ 6 & 6 \end{array} |$

Step 1.6

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

$1 | \begin{array}{c} 12 & - 3 \\ 6 & 6 \end{array} |$

Step 1.7

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

$| \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$

Step 1.8

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

$2 | \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$

Step 1.9

Add the terms together.

$2 | \begin{array}{c} 1 & - 3 \\ - 1 & 6 \end{array} | + 1 | \begin{array}{c} 12 & - 3 \\ 6 & 6 \end{array} | + 2 | \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 1 & - 3 \\ - 1 & 6 \end{array} |$ .

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Step 2.1

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

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

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 $6$ by $1$ .

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

Step 2.2.1.2

Multiply $- - - 3$ .

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Step 2.2.1.2.1

Multiply $- 1$ by $- 3$ .

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

Step 2.2.1.2.2

Multiply $- 1$ by $3$ .

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

Step 2.2.2

Subtract $3$ from $6$ .

$2 \cdot 3 + 1 | \begin{array}{c} 12 & - 3 \\ 6 & 6 \end{array} | + 2 | \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 12 & - 3 \\ 6 & 6 \end{array} |$ .

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Step 3.1

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

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

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 $12$ by $6$ .

$2 \cdot 3 + 1 (72 - 6 \cdot - 3) + 2 | \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$

Step 3.2.1.2

Multiply $- 6$ by $- 3$ .

$2 \cdot 3 + 1 (72 + 18) + 2 | \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$

Step 3.2.2

Add $72$ and $18$ .

$2 \cdot 3 + 1 \cdot 90 + 2 | \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 12 & 1 \\ 6 & - 1 \end{array} |$ .

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Step 4.1

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

$2 \cdot 3 + 1 \cdot 90 + 2 (12 \cdot - 1 - 6 \cdot 1)$

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 $12$ by $- 1$ .

$2 \cdot 3 + 1 \cdot 90 + 2 (- 12 - 6 \cdot 1)$

Step 4.2.1.2

Multiply $- 6$ by $1$ .

$2 \cdot 3 + 1 \cdot 90 + 2 (- 12 - 6)$

Step 4.2.2

Subtract $6$ from $- 12$ .

$2 \cdot 3 + 1 \cdot 90 + 2 \cdot - 18$

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 $3$ .

$6 + 1 \cdot 90 + 2 \cdot - 18$

Step 5.1.2

Multiply $90$ by $1$ .

$6 + 90 + 2 \cdot - 18$

Step 5.1.3

Multiply $2$ by $- 18$ .

$6 + 90 - 36$

Step 5.2

Add $6$ and $90$ .

$96 - 36$

Step 5.3

Subtract $36$ from $96$ .

$60$