Linear Algebra Examples

$[\begin{array}{c} 1 & 4 & - 7 \\ - 1 & - 3 & 4 \\ - 2 & - 4 & 18 \end{array}]$

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.

$| \begin{array}{c} - 3 & 4 \\ - 4 & 18 \end{array} |$

Step 1.4

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

$1 | \begin{array}{c} - 3 & 4 \\ - 4 & 18 \end{array} |$

Step 1.5

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

$| \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} |$

Step 1.6

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

$- 4 | \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} |$

Step 1.7

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

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

Step 1.8

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

$- 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 1.9

Add the terms together.

$1 | \begin{array}{c} - 3 & 4 \\ - 4 & 18 \end{array} | - 4 | \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} | - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} - 3 & 4 \\ - 4 & 18 \end{array} |$ .

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Step 2.1

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

$1 (- 3 \cdot 18 - (- 4 \cdot 4)) - 4 | \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} | - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

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 $- 3$ by $18$ .

$1 (- 54 - (- 4 \cdot 4)) - 4 | \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} | - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 2.2.1.2

Multiply $- (- 4 \cdot 4)$ .

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Step 2.2.1.2.1

Multiply $- 4$ by $4$ .

$1 (- 54 - - 16) - 4 | \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} | - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 2.2.1.2.2

Multiply $- 1$ by $- 16$ .

$1 (- 54 + 16) - 4 | \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} | - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 2.2.2

Add $- 54$ and $16$ .

$1 \cdot - 38 - 4 | \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} | - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} - 1 & 4 \\ - 2 & 18 \end{array} |$ .

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Step 3.1

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

$1 \cdot - 38 - 4 (- 1 \cdot 18 - (- 2 \cdot 4)) - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

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 $- 1$ by $18$ .

$1 \cdot - 38 - 4 (- 18 - (- 2 \cdot 4)) - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 3.2.1.2

Multiply $- (- 2 \cdot 4)$ .

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Step 3.2.1.2.1

Multiply $- 2$ by $4$ .

$1 \cdot - 38 - 4 (- 18 - - 8) - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 3.2.1.2.2

Multiply $- 1$ by $- 8$ .

$1 \cdot - 38 - 4 (- 18 + 8) - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 3.2.2

Add $- 18$ and $8$ .

$1 \cdot - 38 - 4 \cdot - 10 - 7 | \begin{array}{c} - 1 & - 3 \\ - 2 & - 4 \end{array} |$

Step 4

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

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Step 4.1

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

$1 \cdot - 38 - 4 \cdot - 10 - 7 (- - 4 - (- 2 \cdot - 3))$

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 $- 1$ by $- 4$ .

$1 \cdot - 38 - 4 \cdot - 10 - 7 (4 - (- 2 \cdot - 3))$

Step 4.2.1.2

Multiply $- (- 2 \cdot - 3)$ .

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Step 4.2.1.2.1

Multiply $- 2$ by $- 3$ .

$1 \cdot - 38 - 4 \cdot - 10 - 7 (4 - 1 \cdot 6)$

Step 4.2.1.2.2

Multiply $- 1$ by $6$ .

$1 \cdot - 38 - 4 \cdot - 10 - 7 (4 - 6)$

Step 4.2.2

Subtract $6$ from $4$ .

$1 \cdot - 38 - 4 \cdot - 10 - 7 \cdot - 2$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $- 38$ by $1$ .

$- 38 - 4 \cdot - 10 - 7 \cdot - 2$

Step 5.1.2

Multiply $- 4$ by $- 10$ .

$- 38 + 40 - 7 \cdot - 2$

Step 5.1.3

Multiply $- 7$ by $- 2$ .

$- 38 + 40 + 14$

Step 5.2

Add $- 38$ and $40$ .

$2 + 14$

Step 5.3

Add $2$ and $14$ .

$16$