Linear Algebra Examples

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

Step 1

Consider the corresponding sign chart.

$[\begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array}]$

Step 2

Use the sign chart and the given matrix to find the cofactor of each element.

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Step 2.1

Calculate the minor for element $a_{11}$ .

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

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 column $1$ by its cofactor and add.

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Step 2.1.2.1.1

Consider the corresponding sign chart.

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

Step 2.1.2.1.2

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

Step 2.1.2.1.3

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

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

Step 2.1.2.1.4

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

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

Step 2.1.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.1.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.1.2.1.7

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

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

Step 2.1.2.1.8

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

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

Step 2.1.2.1.9

Add the terms together.

$a_{11} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.1.2.2

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

$a_{11} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.1.2.3

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

$a_{11} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0$

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$a_{11} = 1 (1 \cdot 1 + 0 \cdot 2) + 0 + 0$

Step 2.1.2.4.2

Simplify the determinant.

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Step 2.1.2.4.2.1

Simplify each term.

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Step 2.1.2.4.2.1.1

Multiply $1$ by $1$ .

$a_{11} = 1 (1 + 0 \cdot 2) + 0 + 0$

Step 2.1.2.4.2.1.2

Multiply $0$ by $2$ .

$a_{11} = 1 (1 + 0) + 0 + 0$

Step 2.1.2.4.2.2

Add $1$ and $0$ .

$a_{11} = 1 \cdot 1 + 0 + 0$

Step 2.1.2.5

Simplify the determinant.

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Step 2.1.2.5.1

Multiply $1$ by $1$ .

$a_{11} = 1 + 0 + 0$

Step 2.1.2.5.2

Add $1$ and $0$ .

$a_{11} = 1 + 0$

Step 2.1.2.5.3

Add $1$ and $0$ .

$a_{11} = 1$

Step 2.2

Calculate the minor for element $a_{12}$ .

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Step 2.2.1

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

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

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

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 column $1$ by its cofactor and add.

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Step 2.2.2.1.1

Consider the corresponding sign chart.

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

Step 2.2.2.1.2

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

Step 2.2.2.1.3

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

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

Step 2.2.2.1.4

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

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

Step 2.2.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.2.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.2.2.1.7

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

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

Step 2.2.2.1.8

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

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

Step 2.2.2.1.9

Add the terms together.

$a_{12} = 0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.2.2.2

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

$a_{12} = 0 + 0 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.2.2.3

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

$a_{12} = 0 + 0 + 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.2.2.4

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

$a_{12} = 0 + 0 + 0$

Step 2.2.2.5

Simplify the determinant.

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Step 2.2.2.5.1

Add $0$ and $0$ .

$a_{12} = 0 + 0$

Step 2.2.2.5.2

Add $0$ and $0$ .

$a_{12} = 0$

Step 2.3

Calculate the minor for element $a_{13}$ .

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Step 2.3.1

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

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

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

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 column $1$ by its cofactor and add.

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Step 2.3.2.1.1

Consider the corresponding sign chart.

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

Step 2.3.2.1.2

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

Step 2.3.2.1.3

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

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

Step 2.3.2.1.4

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

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

Step 2.3.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.3.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.3.2.1.7

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

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

Step 2.3.2.1.8

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

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

Step 2.3.2.1.9

Add the terms together.

$a_{13} = 0 | \begin{array}{c} 0 & 2 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} |$

Step 2.3.2.2

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

$a_{13} = 0 + 0 | \begin{array}{c} 1 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} |$

Step 2.3.2.3

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

$a_{13} = 0 + 0 + 0 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} |$

Step 2.3.2.4

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

$a_{13} = 0 + 0 + 0$

Step 2.3.2.5

Simplify the determinant.

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Step 2.3.2.5.1

Add $0$ and $0$ .

$a_{13} = 0 + 0$

Step 2.3.2.5.2

Add $0$ and $0$ .

$a_{13} = 0$

Step 2.4

Calculate the minor for element $a_{14}$ .

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Step 2.4.1

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

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

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

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 column $1$ by its cofactor and add.

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Step 2.4.2.1.1

Consider the corresponding sign chart.

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

Step 2.4.2.1.2

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

Step 2.4.2.1.3

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

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

Step 2.4.2.1.4

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

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

Step 2.4.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.4.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.4.2.1.7

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

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

Step 2.4.2.1.8

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

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

Step 2.4.2.1.9

Add the terms together.

$a_{14} = 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 0 & 0 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$

Step 2.4.2.2

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

$a_{14} = 0 + 0 | \begin{array}{c} 1 & 2 \\ 0 & 0 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$

Step 2.4.2.3

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

$a_{14} = 0 + 0 + 0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$

Step 2.4.2.4

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

$a_{14} = 0 + 0 + 0$

Step 2.4.2.5

Simplify the determinant.

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Step 2.4.2.5.1

Add $0$ and $0$ .

$a_{14} = 0 + 0$

Step 2.4.2.5.2

Add $0$ and $0$ .

$a_{14} = 0$

Step 2.5

Calculate the minor for element $a_{21}$ .

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Step 2.5.1

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.5.2

Evaluate the determinant.

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Step 2.5.2.1

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 column $1$ by its cofactor and add.

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Step 2.5.2.1.1

Consider the corresponding sign chart.

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

Step 2.5.2.1.2

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

Step 2.5.2.1.3

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

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

Step 2.5.2.1.4

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

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

Step 2.5.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.5.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.5.2.1.7

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

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

Step 2.5.2.1.8

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

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

Step 2.5.2.1.9

Add the terms together.

$a_{21} = 2 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} |$

Step 2.5.2.2

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

$a_{21} = 2 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} |$

Step 2.5.2.3

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

$a_{21} = 2 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0$

Step 2.5.2.4

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

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Step 2.5.2.4.1

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

$a_{21} = 2 (1 \cdot 1 + 0 \cdot 2) + 0 + 0$

Step 2.5.2.4.2

Simplify the determinant.

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Step 2.5.2.4.2.1

Simplify each term.

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Step 2.5.2.4.2.1.1

Multiply $1$ by $1$ .

$a_{21} = 2 (1 + 0 \cdot 2) + 0 + 0$

Step 2.5.2.4.2.1.2

Multiply $0$ by $2$ .

$a_{21} = 2 (1 + 0) + 0 + 0$

Step 2.5.2.4.2.2

Add $1$ and $0$ .

$a_{21} = 2 \cdot 1 + 0 + 0$

Step 2.5.2.5

Simplify the determinant.

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Step 2.5.2.5.1

Multiply $2$ by $1$ .

$a_{21} = 2 + 0 + 0$

Step 2.5.2.5.2

Add $2$ and $0$ .

$a_{21} = 2 + 0$

Step 2.5.2.5.3

Add $2$ and $0$ .

$a_{21} = 2$

Step 2.6

Calculate the minor for element $a_{22}$ .

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Step 2.6.1

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

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

Step 2.6.2

Evaluate the determinant.

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Step 2.6.2.1

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 column $1$ by its cofactor and add.

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Step 2.6.2.1.1

Consider the corresponding sign chart.

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

Step 2.6.2.1.2

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

Step 2.6.2.1.3

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

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

Step 2.6.2.1.4

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

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

Step 2.6.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.6.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.6.2.1.7

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

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

Step 2.6.2.1.8

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

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

Step 2.6.2.1.9

Add the terms together.

$a_{22} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} |$

Step 2.6.2.2

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

$a_{22} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} |$

Step 2.6.2.3

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

$a_{22} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0$

Step 2.6.2.4

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

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Step 2.6.2.4.1

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

$a_{22} = 1 (1 \cdot 1 + 0 \cdot 2) + 0 + 0$

Step 2.6.2.4.2

Simplify the determinant.

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Step 2.6.2.4.2.1

Simplify each term.

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Step 2.6.2.4.2.1.1

Multiply $1$ by $1$ .

$a_{22} = 1 (1 + 0 \cdot 2) + 0 + 0$

Step 2.6.2.4.2.1.2

Multiply $0$ by $2$ .

$a_{22} = 1 (1 + 0) + 0 + 0$

Step 2.6.2.4.2.2

Add $1$ and $0$ .

$a_{22} = 1 \cdot 1 + 0 + 0$

Step 2.6.2.5

Simplify the determinant.

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Step 2.6.2.5.1

Multiply $1$ by $1$ .

$a_{22} = 1 + 0 + 0$

Step 2.6.2.5.2

Add $1$ and $0$ .

$a_{22} = 1 + 0$

Step 2.6.2.5.3

Add $1$ and $0$ .

$a_{22} = 1$

Step 2.7

Calculate the minor for element $a_{23}$ .

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Step 2.7.1

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

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

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

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 column $1$ by its cofactor and add.

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Step 2.7.2.1.1

Consider the corresponding sign chart.

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

Step 2.7.2.1.2

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

Step 2.7.2.1.3

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

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

Step 2.7.2.1.4

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

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

Step 2.7.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.7.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.7.2.1.7

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

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

Step 2.7.2.1.8

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

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

Step 2.7.2.1.9

Add the terms together.

$a_{23} = 1 | \begin{array}{c} 0 & 2 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 0 & 2 \end{array} |$

Step 2.7.2.2

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

$a_{23} = 1 | \begin{array}{c} 0 & 2 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 2 & 4 \\ 0 & 2 \end{array} |$

Step 2.7.2.3

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

$a_{23} = 1 | \begin{array}{c} 0 & 2 \\ 0 & 1 \end{array} | + 0 + 0$

Step 2.7.2.4

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

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Step 2.7.2.4.1

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

$a_{23} = 1 (0 \cdot 1 + 0 \cdot 2) + 0 + 0$

Step 2.7.2.4.2

Simplify the determinant.

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Step 2.7.2.4.2.1

Simplify each term.

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Step 2.7.2.4.2.1.1

Multiply $0$ by $1$ .

$a_{23} = 1 (0 + 0 \cdot 2) + 0 + 0$

Step 2.7.2.4.2.1.2

Multiply $0$ by $2$ .

$a_{23} = 1 (0 + 0) + 0 + 0$

Step 2.7.2.4.2.2

Add $0$ and $0$ .

$a_{23} = 1 \cdot 0 + 0 + 0$

Step 2.7.2.5

Simplify the determinant.

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Step 2.7.2.5.1

Multiply $0$ by $1$ .

$a_{23} = 0 + 0 + 0$

Step 2.7.2.5.2

Add $0$ and $0$ .

$a_{23} = 0 + 0$

Step 2.7.2.5.3

Add $0$ and $0$ .

$a_{23} = 0$

Step 2.8

Calculate the minor for element $a_{24}$ .

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Step 2.8.1

The minor for $a_{24}$ is the determinant with row $2$ and column $4$ deleted.

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

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

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 2.8.2.1.1

Consider the corresponding sign chart.

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

Step 2.8.2.1.2

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

Step 2.8.2.1.3

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

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

Step 2.8.2.1.4

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

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

Step 2.8.2.1.5

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

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

Step 2.8.2.1.6

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

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

Step 2.8.2.1.7

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

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

Step 2.8.2.1.8

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

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

Step 2.8.2.1.9

Add the terms together.

$a_{24} = 0 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 0 & 0 \end{array} |$

Step 2.8.2.2

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

$a_{24} = 0 + 0 | \begin{array}{c} 1 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 0 & 0 \end{array} |$

Step 2.8.2.3

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

$a_{24} = 0 + 0 + 0 | \begin{array}{c} 1 & 2 \\ 0 & 0 \end{array} |$

Step 2.8.2.4

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

$a_{24} = 0 + 0 + 0$

Step 2.8.2.5

Simplify the determinant.

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Step 2.8.2.5.1

Add $0$ and $0$ .

$a_{24} = 0 + 0$

Step 2.8.2.5.2

Add $0$ and $0$ .

$a_{24} = 0$

Step 2.9

Calculate the minor for element $a_{31}$ .

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Step 2.9.1

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

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

Step 2.9.2

Evaluate the determinant.

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Step 2.9.2.1

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 2.9.2.1.1

Consider the corresponding sign chart.

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

Step 2.9.2.1.2

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

Step 2.9.2.1.3

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

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

Step 2.9.2.1.4

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

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

Step 2.9.2.1.5

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

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

Step 2.9.2.1.6

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

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

Step 2.9.2.1.7

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

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

Step 2.9.2.1.8

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

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

Step 2.9.2.1.9

Add the terms together.

$a_{31} = 0 | \begin{array}{c} 3 & 4 \\ 2 & 3 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 1 & 3 \end{array} | + 1 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.9.2.2

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

$a_{31} = 0 + 0 | \begin{array}{c} 2 & 4 \\ 1 & 3 \end{array} | + 1 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.9.2.3

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

$a_{31} = 0 + 0 + 1 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.9.2.4

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

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Step 2.9.2.4.1

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

$a_{31} = 0 + 0 + 1 (2 \cdot 2 - 1 \cdot 3)$

Step 2.9.2.4.2

Simplify the determinant.

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Step 2.9.2.4.2.1

Simplify each term.

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Step 2.9.2.4.2.1.1

Multiply $2$ by $2$ .

$a_{31} = 0 + 0 + 1 (4 - 1 \cdot 3)$

Step 2.9.2.4.2.1.2

Multiply $- 1$ by $3$ .

$a_{31} = 0 + 0 + 1 (4 - 3)$

Step 2.9.2.4.2.2

Subtract $3$ from $4$ .

$a_{31} = 0 + 0 + 1 \cdot 1$

Step 2.9.2.5

Simplify the determinant.

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Step 2.9.2.5.1

Multiply $1$ by $1$ .

$a_{31} = 0 + 0 + 1$

Step 2.9.2.5.2

Add $0$ and $0$ .

$a_{31} = 0 + 1$

Step 2.9.2.5.3

Add $0$ and $1$ .

$a_{31} = 1$

Step 2.10

Calculate the minor for element $a_{32}$ .

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Step 2.10.1

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

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

Step 2.10.2

Evaluate the determinant.

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Step 2.10.2.1

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 column $1$ by its cofactor and add.

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Step 2.10.2.1.1

Consider the corresponding sign chart.

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

Step 2.10.2.1.2

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

Step 2.10.2.1.3

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

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

Step 2.10.2.1.4

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

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

Step 2.10.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.10.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.10.2.1.7

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

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

Step 2.10.2.1.8

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

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

Step 2.10.2.1.9

Add the terms together.

$a_{32} = 1 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 2 & 3 \end{array} |$

Step 2.10.2.2

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

$a_{32} = 1 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 3 & 4 \\ 2 & 3 \end{array} |$

Step 2.10.2.3

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

$a_{32} = 1 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 + 0$

Step 2.10.2.4

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

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Step 2.10.2.4.1

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

$a_{32} = 1 (2 \cdot 1 + 0 \cdot 3) + 0 + 0$

Step 2.10.2.4.2

Simplify the determinant.

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Step 2.10.2.4.2.1

Simplify each term.

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Step 2.10.2.4.2.1.1

Multiply $2$ by $1$ .

$a_{32} = 1 (2 + 0 \cdot 3) + 0 + 0$

Step 2.10.2.4.2.1.2

Multiply $0$ by $3$ .

$a_{32} = 1 (2 + 0) + 0 + 0$

Step 2.10.2.4.2.2

Add $2$ and $0$ .

$a_{32} = 1 \cdot 2 + 0 + 0$

Step 2.10.2.5

Simplify the determinant.

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Step 2.10.2.5.1

Multiply $2$ by $1$ .

$a_{32} = 2 + 0 + 0$

Step 2.10.2.5.2

Add $2$ and $0$ .

$a_{32} = 2 + 0$

Step 2.10.2.5.3

Add $2$ and $0$ .

$a_{32} = 2$

Step 2.11

Calculate the minor for element $a_{33}$ .

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Step 2.11.1

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

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

Step 2.11.2

Evaluate the determinant.

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Step 2.11.2.1

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 column $1$ by its cofactor and add.

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Step 2.11.2.1.1

Consider the corresponding sign chart.

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

Step 2.11.2.1.2

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

Step 2.11.2.1.3

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

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

Step 2.11.2.1.4

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

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

Step 2.11.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.11.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.11.2.1.7

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

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

Step 2.11.2.1.8

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

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

Step 2.11.2.1.9

Add the terms together.

$a_{33} = 1 | \begin{array}{c} 1 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 1 & 3 \end{array} |$

Step 2.11.2.2

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

$a_{33} = 1 | \begin{array}{c} 1 & 3 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 2 & 4 \\ 1 & 3 \end{array} |$

Step 2.11.2.3

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

$a_{33} = 1 | \begin{array}{c} 1 & 3 \\ 0 & 1 \end{array} | + 0 + 0$

Step 2.11.2.4

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

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Step 2.11.2.4.1

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

$a_{33} = 1 (1 \cdot 1 + 0 \cdot 3) + 0 + 0$

Step 2.11.2.4.2

Simplify the determinant.

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Step 2.11.2.4.2.1

Simplify each term.

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Step 2.11.2.4.2.1.1

Multiply $1$ by $1$ .

$a_{33} = 1 (1 + 0 \cdot 3) + 0 + 0$

Step 2.11.2.4.2.1.2

Multiply $0$ by $3$ .

$a_{33} = 1 (1 + 0) + 0 + 0$

Step 2.11.2.4.2.2

Add $1$ and $0$ .

$a_{33} = 1 \cdot 1 + 0 + 0$

Step 2.11.2.5

Simplify the determinant.

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Step 2.11.2.5.1

Multiply $1$ by $1$ .

$a_{33} = 1 + 0 + 0$

Step 2.11.2.5.2

Add $1$ and $0$ .

$a_{33} = 1 + 0$

Step 2.11.2.5.3

Add $1$ and $0$ .

$a_{33} = 1$

Step 2.12

Calculate the minor for element $a_{34}$ .

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Step 2.12.1

The minor for $a_{34}$ is the determinant with row $3$ and column $4$ deleted.

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

Step 2.12.2

Evaluate the determinant.

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Step 2.12.2.1

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 2.12.2.1.1

Consider the corresponding sign chart.

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

Step 2.12.2.1.2

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

Step 2.12.2.1.3

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

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

Step 2.12.2.1.4

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

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

Step 2.12.2.1.5

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

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

Step 2.12.2.1.6

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

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

Step 2.12.2.1.7

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

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

Step 2.12.2.1.8

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

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

Step 2.12.2.1.9

Add the terms together.

$a_{34} = 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$

Step 2.12.2.2

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

$a_{34} = 0 + 0 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$

Step 2.12.2.3

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

$a_{34} = 0 + 0 + 0 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$

Step 2.12.2.4

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

$a_{34} = 0 + 0 + 0$

Step 2.12.2.5

Simplify the determinant.

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Step 2.12.2.5.1

Add $0$ and $0$ .

$a_{34} = 0 + 0$

Step 2.12.2.5.2

Add $0$ and $0$ .

$a_{34} = 0$

Step 2.13

Calculate the minor for element $a_{41}$ .

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Step 2.13.1

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

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

Step 2.13.2

Evaluate the determinant.

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Step 2.13.2.1

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 column $1$ by its cofactor and add.

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Step 2.13.2.1.1

Consider the corresponding sign chart.

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

Step 2.13.2.1.2

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

Step 2.13.2.1.3

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

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

Step 2.13.2.1.4

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

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

Step 2.13.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.13.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.13.2.1.7

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

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

Step 2.13.2.1.8

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

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

Step 2.13.2.1.9

Add the terms together.

$a_{41} = 2 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} | - 1 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 2 & 3 \end{array} |$

Step 2.13.2.2

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

$a_{41} = 2 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} | - 1 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | + 0$

Step 2.13.2.3

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

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Step 2.13.2.3.1

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

$a_{41} = 2 (2 \cdot 2 - 1 \cdot 3) - 1 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | + 0$

Step 2.13.2.3.2

Simplify the determinant.

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Step 2.13.2.3.2.1

Simplify each term.

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Step 2.13.2.3.2.1.1

Multiply $2$ by $2$ .

$a_{41} = 2 (4 - 1 \cdot 3) - 1 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | + 0$

Step 2.13.2.3.2.1.2

Multiply $- 1$ by $3$ .

$a_{41} = 2 (4 - 3) - 1 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | + 0$

Step 2.13.2.3.2.2

Subtract $3$ from $4$ .

$a_{41} = 2 \cdot 1 - 1 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | + 0$

Step 2.13.2.4

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

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Step 2.13.2.4.1

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

$a_{41} = 2 \cdot 1 - 1 (3 \cdot 2 - 1 \cdot 4) + 0$

Step 2.13.2.4.2

Simplify the determinant.

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Step 2.13.2.4.2.1

Simplify each term.

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Step 2.13.2.4.2.1.1

Multiply $3$ by $2$ .

$a_{41} = 2 \cdot 1 - 1 (6 - 1 \cdot 4) + 0$

Step 2.13.2.4.2.1.2

Multiply $- 1$ by $4$ .

$a_{41} = 2 \cdot 1 - 1 (6 - 4) + 0$

Step 2.13.2.4.2.2

Subtract $4$ from $6$ .

$a_{41} = 2 \cdot 1 - 1 \cdot 2 + 0$

Step 2.13.2.5

Simplify the determinant.

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Step 2.13.2.5.1

Simplify each term.

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Step 2.13.2.5.1.1

Multiply $2$ by $1$ .

$a_{41} = 2 - 1 \cdot 2 + 0$

Step 2.13.2.5.1.2

Multiply $- 1$ by $2$ .

$a_{41} = 2 - 2 + 0$

Step 2.13.2.5.2

Subtract $2$ from $2$ .

$a_{41} = 0 + 0$

Step 2.13.2.5.3

Add $0$ and $0$ .

$a_{41} = 0$

Step 2.14

Calculate the minor for element $a_{42}$ .

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Step 2.14.1

The minor for $a_{42}$ is the determinant with row $4$ and column $2$ deleted.

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

Step 2.14.2

Evaluate the determinant.

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Step 2.14.2.1

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 column $1$ by its cofactor and add.

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Step 2.14.2.1.1

Consider the corresponding sign chart.

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

Step 2.14.2.1.2

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

Step 2.14.2.1.3

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

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

Step 2.14.2.1.4

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

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

Step 2.14.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.14.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.14.2.1.7

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

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

Step 2.14.2.1.8

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

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

Step 2.14.2.1.9

Add the terms together.

$a_{42} = 1 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 4 \\ 2 & 3 \end{array} |$

Step 2.14.2.2

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

$a_{42} = 1 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} | + 0 + 0 | \begin{array}{c} 3 & 4 \\ 2 & 3 \end{array} |$

Step 2.14.2.3

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

$a_{42} = 1 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} | + 0 + 0$

Step 2.14.2.4

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

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Step 2.14.2.4.1

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

$a_{42} = 1 (2 \cdot 2 - 1 \cdot 3) + 0 + 0$

Step 2.14.2.4.2

Simplify the determinant.

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Step 2.14.2.4.2.1

Simplify each term.

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Step 2.14.2.4.2.1.1

Multiply $2$ by $2$ .

$a_{42} = 1 (4 - 1 \cdot 3) + 0 + 0$

Step 2.14.2.4.2.1.2

Multiply $- 1$ by $3$ .

$a_{42} = 1 (4 - 3) + 0 + 0$

Step 2.14.2.4.2.2

Subtract $3$ from $4$ .

$a_{42} = 1 \cdot 1 + 0 + 0$

Step 2.14.2.5

Simplify the determinant.

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Step 2.14.2.5.1

Multiply $1$ by $1$ .

$a_{42} = 1 + 0 + 0$

Step 2.14.2.5.2

Add $1$ and $0$ .

$a_{42} = 1 + 0$

Step 2.14.2.5.3

Add $1$ and $0$ .

$a_{42} = 1$

Step 2.15

Calculate the minor for element $a_{43}$ .

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Step 2.15.1

The minor for $a_{43}$ is the determinant with row $4$ and column $3$ deleted.

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

Step 2.15.2

Evaluate the determinant.

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Step 2.15.2.1

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 column $1$ by its cofactor and add.

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Step 2.15.2.1.1

Consider the corresponding sign chart.

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

Step 2.15.2.1.2

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

Step 2.15.2.1.3

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

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

Step 2.15.2.1.4

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

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

Step 2.15.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.15.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.15.2.1.7

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

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

Step 2.15.2.1.8

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

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

Step 2.15.2.1.9

Add the terms together.

$a_{43} = 1 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 0 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 1 & 3 \end{array} |$

Step 2.15.2.2

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

$a_{43} = 1 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} | + 0 + 0 | \begin{array}{c} 2 & 4 \\ 1 & 3 \end{array} |$

Step 2.15.2.3

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

$a_{43} = 1 | \begin{array}{c} 1 & 3 \\ 0 & 2 \end{array} | + 0 + 0$

Step 2.15.2.4

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

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Step 2.15.2.4.1

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

$a_{43} = 1 (1 \cdot 2 + 0 \cdot 3) + 0 + 0$

Step 2.15.2.4.2

Simplify the determinant.

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Step 2.15.2.4.2.1

Simplify each term.

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Step 2.15.2.4.2.1.1

Multiply $2$ by $1$ .

$a_{43} = 1 (2 + 0 \cdot 3) + 0 + 0$

Step 2.15.2.4.2.1.2

Multiply $0$ by $3$ .

$a_{43} = 1 (2 + 0) + 0 + 0$

Step 2.15.2.4.2.2

Add $2$ and $0$ .

$a_{43} = 1 \cdot 2 + 0 + 0$

Step 2.15.2.5

Simplify the determinant.

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Step 2.15.2.5.1

Multiply $2$ by $1$ .

$a_{43} = 2 + 0 + 0$

Step 2.15.2.5.2

Add $2$ and $0$ .

$a_{43} = 2 + 0$

Step 2.15.2.5.3

Add $2$ and $0$ .

$a_{43} = 2$

Step 2.16

Calculate the minor for element $a_{44}$ .

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Step 2.16.1

The minor for $a_{44}$ is the determinant with row $4$ and column $4$ deleted.

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

Step 2.16.2

Evaluate the determinant.

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Step 2.16.2.1

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 column $1$ by its cofactor and add.

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Step 2.16.2.1.1

Consider the corresponding sign chart.

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

Step 2.16.2.1.2

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

Step 2.16.2.1.3

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

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

Step 2.16.2.1.4

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

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

Step 2.16.2.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 2.16.2.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 2.16.2.1.7

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

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

Step 2.16.2.1.8

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

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

Step 2.16.2.1.9

Add the terms together.

$a_{44} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.16.2.2

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

$a_{44} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 2 & 3 \\ 1 & 2 \end{array} |$

Step 2.16.2.3

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

$a_{44} = 1 | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | + 0 + 0$

Step 2.16.2.4

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

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Step 2.16.2.4.1

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

$a_{44} = 1 (1 \cdot 1 + 0 \cdot 2) + 0 + 0$

Step 2.16.2.4.2

Simplify the determinant.

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Step 2.16.2.4.2.1

Simplify each term.

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Step 2.16.2.4.2.1.1

Multiply $1$ by $1$ .

$a_{44} = 1 (1 + 0 \cdot 2) + 0 + 0$

Step 2.16.2.4.2.1.2

Multiply $0$ by $2$ .

$a_{44} = 1 (1 + 0) + 0 + 0$

Step 2.16.2.4.2.2

Add $1$ and $0$ .

$a_{44} = 1 \cdot 1 + 0 + 0$

Step 2.16.2.5

Simplify the determinant.

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Step 2.16.2.5.1

Multiply $1$ by $1$ .

$a_{44} = 1 + 0 + 0$

Step 2.16.2.5.2

Add $1$ and $0$ .

$a_{44} = 1 + 0$

Step 2.16.2.5.3

Add $1$ and $0$ .

$a_{44} = 1$

Step 2.17

The cofactor matrix is a matrix of the minors with the sign changed for the elements in the $-$ positions on the sign chart.

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

Step 3

Transpose the matrix by switching its rows to columns.

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