Linear Algebra Examples

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

Step 1

Find the determinant.

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Step 1.1

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 1.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 1.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

Add the terms together.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

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 $3$ by its cofactor and add.

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Step 1.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.5.1.3

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

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

Step 1.5.1.4

Multiply element $a_{31}$ by its cofactor.

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

Step 1.5.1.5

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

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

Step 1.5.1.6

Multiply element $a_{32}$ by its cofactor.

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

Step 1.5.1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

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

Step 1.5.1.8

Multiply element $a_{33}$ by its cofactor.

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

Step 1.5.1.9

Add the terms together.

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

Simplify the determinant.

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Step 1.5.5.1

Add $0$ and $0$ .

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

Step 1.5.5.2

Add $0$ and $0$ .

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

Step 1.6

Simplify the determinant.

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Step 1.6.1

Add $0$ and $0$ .

$0 + 0 - 1 \cdot 0$

Step 1.6.2

Add $0$ and $0$ .

$0 - 1 \cdot 0$

Step 1.6.3

Subtract $0$ from $0$ .

$0$

Step 2

There is no inverse because the determinant is $0$ .