Linear Algebra Examples

$[\begin{array}{c} 2 & \sqrt{5} \\ \sqrt{5} & 5 \end{array}]$

Step 1

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

$2 \cdot 5 - \sqrt{5} \sqrt{5}$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2$ by $5$ .

$10 - \sqrt{5} \sqrt{5}$

Step 2.1.2

Multiply $- \sqrt{5} \sqrt{5}$ .

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Step 2.1.2.1

Raise $\sqrt{5}$ to the power of $1$ .

$10 - ({\sqrt{5}}^{1} \sqrt{5})$

Step 2.1.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$10 - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 2.1.2.3

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

$10 - {\sqrt{5}}^{1 + 1}$

Step 2.1.2.4

Add $1$ and $1$ .

$10 - {\sqrt{5}}^{2}$

Step 2.1.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$10 - {(5^{\frac{1}{2}})}^{2}$

Step 2.1.3.2

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

$10 - 5^{\frac{1}{2} \cdot 2}$

Step 2.1.3.3

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

$10 - 5^{\frac{2}{2}}$

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Cancel the common factor.

$10 - 5^{\frac{2}{2}}$

Step 2.1.3.4.2

Rewrite the expression.

$10 - 5^{1}$

Step 2.1.3.5

Evaluate the exponent.

$10 - 1 \cdot 5$

Step 2.1.4

Multiply $- 1$ by $5$ .

$10 - 5$

Step 2.2

Subtract $5$ from $10$ .

$5$