Linear Algebra Examples

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

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Add the terms together.

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

Step 2

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

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

Step 3

Multiply $0$ by $| \begin{array}{c} 4 & 3 & 3 \\ 1 & 2 & 4 \\ 1 & 2 & 4 \end{array} |$ .

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

Step 4

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

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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.

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.1.7

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

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

Step 4.1.8

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

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

Step 4.1.9

Add the terms together.

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

Step 4.2

Multiply $0$ by $| \begin{array}{c} 1 & 4 \\ 1 & 4 \end{array} |$ .

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

Step 4.3

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

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Step 4.3.1

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

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

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 $4$ .

$0 - 3 (4 (8 - 3 \cdot 4) + 0 + 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |) + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Step 4.3.2.1.2

Multiply $- 3$ by $4$ .

$0 - 3 (4 (8 - 12) + 0 + 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |) + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Step 4.3.2.2

Subtract $12$ from $8$ .

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

Step 4.4

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

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Step 4.4.1

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

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

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 $3$ by $1$ .

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

Step 4.4.2.1.2

Multiply $- 1$ by $2$ .

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

Step 4.4.2.2

Subtract $2$ from $3$ .

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

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 $4$ by $- 4$ .

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

Step 4.5.1.2

Multiply $3$ by $1$ .

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

Step 4.5.2

Add $- 16$ and $0$ .

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

Step 4.5.3

Add $- 16$ and $3$ .

$0 - 3 \cdot - 13 + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Step 5

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

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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.

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.1.7

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

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

Step 5.1.8

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

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

Step 5.1.9

Add the terms together.

$0 - 3 \cdot - 13 + 0 - 1 (4 | \begin{array}{c} 2 & 2 \\ 2 & 3 \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 1 & 2 \end{array} |)$

Step 5.2

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

$0 - 3 \cdot - 13 + 0 - 1 (4 | \begin{array}{c} 2 & 2 \\ 2 & 3 \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Step 5.3

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

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Step 5.3.1

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

$0 - 3 \cdot - 13 + 0 - 1 (4 (2 \cdot 3 - 2 \cdot 2) - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

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 $2$ by $3$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 (6 - 2 \cdot 2) - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Step 5.3.2.1.2

Multiply $- 2$ by $2$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 (6 - 4) - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Step 5.3.2.2

Subtract $4$ from $6$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Step 5.4

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

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Step 5.4.1

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

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 (1 \cdot 3 - 1 \cdot 2) + 0)$

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 $1$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 (3 - 1 \cdot 2) + 0)$

Step 5.4.2.1.2

Multiply $- 1$ by $2$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 (3 - 2) + 0)$

Step 5.4.2.2

Subtract $2$ from $3$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 \cdot 1 + 0)$

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 $4$ by $2$ .

$0 - 3 \cdot - 13 + 0 - 1 (8 - 3 \cdot 1 + 0)$

Step 5.5.1.2

Multiply $- 3$ by $1$ .

$0 - 3 \cdot - 13 + 0 - 1 (8 - 3 + 0)$

Step 5.5.2

Subtract $3$ from $8$ .

$0 - 3 \cdot - 13 + 0 - 1 (5 + 0)$

Step 5.5.3

Add $5$ and $0$ .

$0 - 3 \cdot - 13 + 0 - 1 \cdot 5$

Step 6

Simplify the determinant.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $- 3$ by $- 13$ .

$0 + 39 + 0 - 1 \cdot 5$

Step 6.1.2

Multiply $- 1$ by $5$ .

$0 + 39 + 0 - 5$

Step 6.2

Add $0$ and $39$ .

$39 + 0 - 5$

Step 6.3

Add $39$ and $0$ .

$39 - 5$

Step 6.4

Subtract $5$ from $39$ .

$34$

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