Linear Algebra Examples

$[\begin{array}{c} 3 & 0 & 1 \\ 2 & 4 & 4 \\ 4 & 4 & 3 \end{array}]$

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

$0 | \begin{array}{c} 2 & 4 \\ 4 & 3 \end{array} |$

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Add the terms together.

$3 | \begin{array}{c} 4 & 4 \\ 4 & 3 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 4 & 3 \end{array} | + 1 | \begin{array}{c} 2 & 4 \\ 4 & 4 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 2 & 4 \\ 4 & 3 \end{array} |$ .

$3 | \begin{array}{c} 4 & 4 \\ 4 & 3 \end{array} | + 0 + 1 | \begin{array}{c} 2 & 4 \\ 4 & 4 \end{array} |$

Step 3

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

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Step 3.1

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

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

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 $4$ by $3$ .

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

Step 3.2.1.2

Multiply $- 4$ by $4$ .

$3 (12 - 16) + 0 + 1 | \begin{array}{c} 2 & 4 \\ 4 & 4 \end{array} |$

Step 3.2.2

Subtract $16$ from $12$ .

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

Step 4

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

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Step 4.1

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

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

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 $4$ .

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

Step 4.2.1.2

Multiply $- 4$ by $4$ .

$3 \cdot - 4 + 0 + 1 (8 - 16)$

Step 4.2.2

Subtract $16$ from $8$ .

$3 \cdot - 4 + 0 + 1 \cdot - 8$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $3$ by $- 4$ .

$- 12 + 0 + 1 \cdot - 8$

Step 5.1.2

Multiply $- 8$ by $1$ .

$- 12 + 0 - 8$

Step 5.2

Add $- 12$ and $0$ .

$- 12 - 8$

Step 5.3

Subtract $8$ from $- 12$ .

$- 20$

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