Linear Algebra Examples

$[\begin{array}{c} x & x^{2} & \frac{1}{x} \\ 1 & 2 x & - \frac{1}{x^{2}} \\ 0 & 2 & \frac{2}{x^{3}} \end{array}]$

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.

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

Step 1.4

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

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

Step 1.5

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

$| \begin{array}{c} 1 & - \frac{1}{x^{2}} \\ 0 & \frac{2}{x^{3}} \end{array} |$

Step 1.6

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

$- x^{2} | \begin{array}{c} 1 & - \frac{1}{x^{2}} \\ 0 & \frac{2}{x^{3}} \end{array} |$

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Add the terms together.

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

Step 2

Evaluate $| \begin{array}{c} 2 x & - \frac{1}{x^{2}} \\ 2 & \frac{2}{x^{3}} \end{array} |$ .

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Step 2.1

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

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

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $x$ .

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Step 2.2.1.1.1

Factor $x$ out of $2 x$ .

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

Step 2.2.1.1.2

Factor $x$ out of $x^{3}$ .

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

Step 2.2.1.1.3

Cancel the common factor.

Step 2.2.1.1.4

Rewrite the expression.

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

Step 2.2.1.2

Combine $2$ and $\frac{2}{x^{2}}$ .

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

Step 2.2.1.3

Multiply $2$ by $2$ .

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

Step 2.2.1.4

Multiply $- 2 (- \frac{1}{x^{2}})$ .

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Step 2.2.1.4.1

Multiply $- 1$ by $- 2$ .

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

Step 2.2.1.4.2

Combine $2$ and $\frac{1}{x^{2}}$ .

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

Step 2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.3

Add $4$ and $2$ .

$x \frac{6}{x^{2}} - x^{2} | \begin{array}{c} 1 & - \frac{1}{x^{2}} \\ 0 & \frac{2}{x^{3}} \end{array} | + \frac{1}{x} | \begin{array}{c} 1 & 2 x \\ 0 & 2 \end{array} |$

Step 3

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

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Step 3.1

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

$x \frac{6}{x^{2}} - x^{2} (1 \frac{2}{x^{3}} + 0 (- \frac{1}{x^{2}})) + \frac{1}{x} | \begin{array}{c} 1 & 2 x \\ 0 & 2 \end{array} |$

Step 3.2

Simplify the determinant.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $\frac{2}{x^{3}}$ by $1$ .

$x \frac{6}{x^{2}} - x^{2} (\frac{2}{x^{3}} + 0 (- \frac{1}{x^{2}})) + \frac{1}{x} | \begin{array}{c} 1 & 2 x \\ 0 & 2 \end{array} |$

Step 3.2.1.2

Multiply $0 (- \frac{1}{x^{2}})$ .

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Step 3.2.1.2.1

Multiply $- 1$ by $0$ .

$x \frac{6}{x^{2}} - x^{2} (\frac{2}{x^{3}} + 0 \frac{1}{x^{2}}) + \frac{1}{x} | \begin{array}{c} 1 & 2 x \\ 0 & 2 \end{array} |$

Step 3.2.1.2.2

Multiply $0$ by $\frac{1}{x^{2}}$ .

$x \frac{6}{x^{2}} - x^{2} (\frac{2}{x^{3}} + 0) + \frac{1}{x} | \begin{array}{c} 1 & 2 x \\ 0 & 2 \end{array} |$

Step 3.2.2

Add $\frac{2}{x^{3}}$ and $0$ .

$x \frac{6}{x^{2}} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} | \begin{array}{c} 1 & 2 x \\ 0 & 2 \end{array} |$

Step 4

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

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Step 4.1

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

$x \frac{6}{x^{2}} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} (1 \cdot 2 + 0 (2 x))$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $2$ by $1$ .

$x \frac{6}{x^{2}} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} (2 + 0 (2 x))$

Step 4.2.1.2

Multiply $0 (2 x)$ .

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Step 4.2.1.2.1

Multiply $2$ by $0$ .

$x \frac{6}{x^{2}} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} (2 + 0 x)$

Step 4.2.1.2.2

Multiply $0$ by $x$ .

$x \frac{6}{x^{2}} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} (2 + 0)$

Step 4.2.2

Add $2$ and $0$ .

$x \frac{6}{x^{2}} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} \cdot 2$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Cancel the common factor of $x$ .

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Step 5.1.1.1

Factor $x$ out of $x^{2}$ .

$x \frac{6}{x \cdot x} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} \cdot 2$

Step 5.1.1.2

Cancel the common factor.

$x \frac{6}{x \cdot x} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} \cdot 2$

Step 5.1.1.3

Rewrite the expression.

$\frac{6}{x} - x^{2} \frac{2}{x^{3}} + \frac{1}{x} \cdot 2$

Step 5.1.2

Cancel the common factor of $x^{2}$ .

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Step 5.1.2.1

Factor $x^{2}$ out of $- x^{2}$ .

$\frac{6}{x} + x^{2} \cdot - 1 \frac{2}{x^{3}} + \frac{1}{x} \cdot 2$

Step 5.1.2.2

Factor $x^{2}$ out of $x^{3}$ .

$\frac{6}{x} + x^{2} \cdot - 1 \frac{2}{x^{2} x} + \frac{1}{x} \cdot 2$

Step 5.1.2.3

Cancel the common factor.

$\frac{6}{x} + x^{2} \cdot - 1 \frac{2}{x^{2} x} + \frac{1}{x} \cdot 2$

Step 5.1.2.4

Rewrite the expression.

$\frac{6}{x} - 1 \frac{2}{x} + \frac{1}{x} \cdot 2$

Step 5.1.3

Rewrite $- 1 \frac{2}{x}$ as $- \frac{2}{x}$ .

$\frac{6}{x} - \frac{2}{x} + \frac{1}{x} \cdot 2$

Step 5.1.4

Combine $\frac{1}{x}$ and $2$ .

$\frac{6}{x} - \frac{2}{x} + \frac{2}{x}$

Step 5.2

Combine the opposite terms in $\frac{6}{x} - \frac{2}{x} + \frac{2}{x}$ .

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Step 5.2.1

Add $- \frac{2}{x}$ and $\frac{2}{x}$ .

$\frac{6}{x} + 0$

Step 5.2.2

Add $\frac{6}{x}$ and $0$ .

$\frac{6}{x}$