Linear Algebra Examples

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

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Add the terms together.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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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.

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.1.7

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

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

Step 5.1.8

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

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

Step 5.1.9

Add the terms together.

$1 (1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |) + 0 + 0 + 0$

Step 5.2

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

$1 (1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |) + 0 + 0 + 0$

Step 5.3

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

$1 (1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 + 0) + 0 + 0 + 0$

Step 5.4

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

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Step 5.4.1

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

$1 (1 (1 \cdot 1 + 0 \cdot 0) + 0 + 0) + 0 + 0 + 0$

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 $1$ by $1$ .

$1 (1 (1 + 0 \cdot 0) + 0 + 0) + 0 + 0 + 0$

Step 5.4.2.1.2

Multiply $0$ by $0$ .

$1 (1 (1 + 0) + 0 + 0) + 0 + 0 + 0$

Step 5.4.2.2

Add $1$ and $0$ .

$1 (1 \cdot 1 + 0 + 0) + 0 + 0 + 0$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Multiply $1$ by $1$ .

$1 (1 + 0 + 0) + 0 + 0 + 0$

Step 5.5.2

Add $1$ and $0$ .

$1 (1 + 0) + 0 + 0 + 0$

Step 5.5.3

Add $1$ and $0$ .

$1 \cdot 1 + 0 + 0 + 0$

Step 6

Simplify the determinant.

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Step 6.1

Multiply $1$ by $1$ .

$1 + 0 + 0 + 0$

Step 6.2

Add $1$ and $0$ .

$1 + 0 + 0$

Step 6.3

Add $1$ and $0$ .

$1 + 0$

Step 6.4

Add $1$ and $0$ .

$1$