Linear Algebra Examples

$[\begin{array}{c} 3 & 1 & 5 \\ 2 & i & 7 \\ i & k^{3} & 8 \end{array}]$

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.

$| \begin{array}{c} i & 7 \\ k^{3} & 8 \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

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

$a_{11} = i \cdot 8 - k^{3} \cdot 7$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Move $8$ to the left of $i$ .

$a_{11} = 8 \cdot i - k^{3} \cdot 7$

Step 2.1.2.2.2

Multiply $7$ by $- 1$ .

$a_{11} = 8 i - 7 k^{3}$

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.

$| \begin{array}{c} 2 & 7 \\ i & 8 \end{array} |$

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

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

$a_{12} = 2 \cdot 8 - i \cdot 7$

Step 2.2.2.2

Simplify each term.

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Step 2.2.2.2.1

Multiply $2$ by $8$ .

$a_{12} = 16 - i \cdot 7$

Step 2.2.2.2.2

Multiply $7$ by $- 1$ .

$a_{12} = 16 - 7 i$

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.

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

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

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

$a_{13} = 2 k^{3} - i i$

Step 2.3.2.2

Simplify each term.

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Step 2.3.2.2.1

Multiply $- i i$ .

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Step 2.3.2.2.1.1

Raise $i$ to the power of $1$ .

$a_{13} = 2 k^{3} - (i^{1} i)$

Step 2.3.2.2.1.2

Raise $i$ to the power of $1$ .

$a_{13} = 2 k^{3} - (i^{1} i^{1})$

Step 2.3.2.2.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{13} = 2 k^{3} - i^{1 + 1}$

Step 2.3.2.2.1.4

Add $1$ and $1$ .

$a_{13} = 2 k^{3} - i^{2}$

Step 2.3.2.2.2

Rewrite $i^{2}$ as $- 1$ .

$a_{13} = 2 k^{3} - - 1$

Step 2.3.2.2.3

Multiply $- 1$ by $- 1$ .

$a_{13} = 2 k^{3} + 1$

Step 2.4

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

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Step 2.4.1

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

$| \begin{array}{c} 1 & 5 \\ k^{3} & 8 \end{array} |$

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

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

$a_{21} = 1 \cdot 8 - k^{3} \cdot 5$

Step 2.4.2.2

Simplify each term.

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Step 2.4.2.2.1

Multiply $8$ by $1$ .

$a_{21} = 8 - k^{3} \cdot 5$

Step 2.4.2.2.2

Multiply $5$ by $- 1$ .

$a_{21} = 8 - 5 k^{3}$

Step 2.5

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

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Step 2.5.1

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

$| \begin{array}{c} 3 & 5 \\ i & 8 \end{array} |$

Step 2.5.2

Evaluate the determinant.

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Step 2.5.2.1

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

$a_{22} = 3 \cdot 8 - i \cdot 5$

Step 2.5.2.2

Simplify each term.

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Step 2.5.2.2.1

Multiply $3$ by $8$ .

$a_{22} = 24 - i \cdot 5$

Step 2.5.2.2.2

Multiply $5$ by $- 1$ .

$a_{22} = 24 - 5 i$

Step 2.6

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

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Step 2.6.1

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

$| \begin{array}{c} 3 & 1 \\ i & k^{3} \end{array} |$

Step 2.6.2

Evaluate the determinant.

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Step 2.6.2.1

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

$a_{23} = 3 k^{3} - i \cdot 1$

Step 2.6.2.2

Multiply $- 1$ by $1$ .

$a_{23} = 3 k^{3} - i$

Step 2.7

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

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Step 2.7.1

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

$| \begin{array}{c} 1 & 5 \\ i & 7 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

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

$a_{31} = 1 \cdot 7 - i \cdot 5$

Step 2.7.2.2

Simplify each term.

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Step 2.7.2.2.1

Multiply $7$ by $1$ .

$a_{31} = 7 - i \cdot 5$

Step 2.7.2.2.2

Multiply $5$ by $- 1$ .

$a_{31} = 7 - 5 i$

Step 2.8

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

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Step 2.8.1

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

$| \begin{array}{c} 3 & 5 \\ 2 & 7 \end{array} |$

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

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

$a_{32} = 3 \cdot 7 - 2 \cdot 5$

Step 2.8.2.2

Simplify the determinant.

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Step 2.8.2.2.1

Simplify each term.

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Step 2.8.2.2.1.1

Multiply $3$ by $7$ .

$a_{32} = 21 - 2 \cdot 5$

Step 2.8.2.2.1.2

Multiply $- 2$ by $5$ .

$a_{32} = 21 - 10$

Step 2.8.2.2.2

Subtract $10$ from $21$ .

$a_{32} = 11$

Step 2.9

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

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Step 2.9.1

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

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

Step 2.9.2

Evaluate the determinant.

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Step 2.9.2.1

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

$a_{33} = 3 i - 2 \cdot 1$

Step 2.9.2.2

Simplify the determinant.

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Step 2.9.2.2.1

Multiply $- 2$ by $1$ .

$a_{33} = 3 i - 2$

Step 2.9.2.2.2

Reorder $3 i$ and $- 2$ .

$a_{33} = - 2 + 3 i$

Step 2.10

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

$[\begin{array}{c} 8 i - 7 k^{3} & - (16 - 7 i) & 2 k^{3} + 1 \\ - (8 - 5 k^{3}) & 24 - 5 i & - (3 k^{3} - i) \\ 7 - 5 i & - 11 & - 2 + 3 i \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 8 i - 7 k^{3} & - (8 - 5 k^{3}) & 7 - 5 i \\ - (16 - 7 i) & 24 - 5 i & - 11 \\ 2 k^{3} + 1 & - (3 k^{3} - i) & - 2 + 3 i \end{array}]$