Linear Algebra Examples

$A = [\begin{array}{c} 3 \sqrt{3} & - 3 \\ 3 & 3 \sqrt{3} \end{array}]$

Step 1

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

$3 \sqrt{3} (3 \sqrt{3}) - 3 \cdot - 3$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $3 \sqrt{3} (3 \sqrt{3})$ .

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Step 2.1.1.1

Multiply $3$ by $3$ .

$9 \sqrt{3} \sqrt{3} - 3 \cdot - 3$

Step 2.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$9 ({\sqrt{3}}^{1} \sqrt{3}) - 3 \cdot - 3$

Step 2.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$9 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 3 \cdot - 3$

Step 2.1.1.4

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

$9 {\sqrt{3}}^{1 + 1} - 3 \cdot - 3$

Step 2.1.1.5

Add $1$ and $1$ .

$9 {\sqrt{3}}^{2} - 3 \cdot - 3$

Step 2.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$9 {(3^{\frac{1}{2}})}^{2} - 3 \cdot - 3$

Step 2.1.2.2

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

$9 \cdot 3^{\frac{1}{2} \cdot 2} - 3 \cdot - 3$

Step 2.1.2.3

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

$9 \cdot 3^{\frac{2}{2}} - 3 \cdot - 3$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$9 \cdot 3^{\frac{2}{2}} - 3 \cdot - 3$

Step 2.1.2.4.2

Rewrite the expression.

$9 \cdot 3^{1} - 3 \cdot - 3$

Step 2.1.2.5

Evaluate the exponent.

$9 \cdot 3 - 3 \cdot - 3$

Step 2.1.3

Multiply $9$ by $3$ .

$27 - 3 \cdot - 3$

Step 2.1.4

Multiply $- 3$ by $- 3$ .

$27 + 9$

Step 2.2

Add $27$ and $9$ .

$36$