Linear Algebra Examples

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{1}{\sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 1

Multiply $\frac{1}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{1}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{\sqrt{22} \sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.2

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.3

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.4

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{1 + 1}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.5

Add $1$ and $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{2}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.6

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22^{\frac{1}{2} \cdot 2}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.6.3

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22^{\frac{2}{2}}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

Step 2.6.4.2

Rewrite the expression.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22^{1}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 2.6.5

Evaluate the exponent.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 3

Multiply $\frac{2}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{2}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{\sqrt{22} \sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.2

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.3

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.4

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{1 + 1}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.5

Add $1$ and $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{2}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.6

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22^{\frac{1}{2} \cdot 2}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.6.3

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22^{\frac{2}{2}}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

Step 4.6.4.2

Rewrite the expression.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22^{1}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 4.6.5

Evaluate the exponent.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 5

Cancel the common factor of $2$ and $22$ .

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Step 5.1

Factor $2$ out of $2 \sqrt{22}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 (\sqrt{22})}{22} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $22$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{2 \cdot 11} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 5.2.2

Cancel the common factor.

Step 5.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 6

Multiply $\frac{4}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - (\frac{4}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}}) & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{4}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{\sqrt{22} \sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.2

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.3

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.4

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{1 + 1}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.5

Add $1$ and $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{2}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.6

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22^{\frac{1}{2} \cdot 2}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.6.3

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22^{\frac{2}{2}}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

Step 7.6.4.2

Rewrite the expression.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22^{1}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 7.6.5

Evaluate the exponent.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 8

Cancel the common factor of $4$ and $22$ .

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Step 8.1

Factor $2$ out of $4 \sqrt{22}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 (2 \sqrt{22})}{22} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $22$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 (2 \sqrt{22})}{2 (11)} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 8.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 (2 \sqrt{22})}{2 \cdot 11} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 8.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 9

Multiply $\frac{1}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{1}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{\sqrt{22} \sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.2

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.3

Raise $\sqrt{22}$ to the power of $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.4

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{1 + 1}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.5

Add $1$ and $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{2}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.6

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

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Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{\frac{1}{2} \cdot 2}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.6.3

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

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{\frac{2}{2}}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{\frac{2}{2}}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.6.4.2

Rewrite the expression.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{1}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 10.6.5

Evaluate the exponent.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Step 11

Multiply $[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$ .

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Step 11.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 4$ and the second matrix is $4 \times 2$ .

Step 11.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} \frac{2}{7} \cdot 2 + \frac{5}{7} \cdot 5 + \frac{2}{7} \cdot 2 - \frac{4}{7} \cdot - 4 & \frac{2}{7} \cdot 3 + \frac{5}{7} \cdot 7 + \frac{2}{7} \cdot - 2 - \frac{4}{7} \cdot - 3 \\ \frac{\sqrt{22}}{22} \cdot 2 + \frac{\sqrt{22}}{11} \cdot 5 - \frac{2 \sqrt{22}}{11} \cdot 2 + \frac{\sqrt{22}}{22} \cdot - 4 & \frac{\sqrt{22}}{22} \cdot 3 + \frac{\sqrt{22}}{11} \cdot 7 - \frac{2 \sqrt{22}}{11} \cdot - 2 + \frac{\sqrt{22}}{22} \cdot - 3 \end{array}]$

Step 11.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 7 & 7 \\ 0 & \sqrt{22} \end{array}]$