Linear Algebra Examples

$[\begin{array}{c} \sqrt{2} & 2 \sqrt{3} \\ \sqrt{6} & 5 \end{array}]$

Step 1

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

$\sqrt{2} \cdot 5 - \sqrt{6} (2 \sqrt{3})$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Move $5$ to the left of $\sqrt{2}$ .

$5 \cdot \sqrt{2} - \sqrt{6} (2 \sqrt{3})$

Step 2.1.2

Multiply $- \sqrt{6} (2 \sqrt{3})$ .

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Step 2.1.2.1

Multiply $2$ by $- 1$ .

$5 \sqrt{2} - 2 \sqrt{6} \sqrt{3}$

Step 2.1.2.2

Combine using the product rule for radicals.

$5 \sqrt{2} - 2 \sqrt{3 \cdot 6}$

Step 2.1.2.3

Multiply $3$ by $6$ .

$5 \sqrt{2} - 2 \sqrt{18}$

Step 2.1.3

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.3.1

Factor $9$ out of $18$ .

$5 \sqrt{2} - 2 \sqrt{9 (2)}$

Step 2.1.3.2

Rewrite $9$ as $3^{2}$ .

$5 \sqrt{2} - 2 \sqrt{3^{2} \cdot 2}$

Step 2.1.4

Pull terms out from under the radical.

$5 \sqrt{2} - 2 (3 \sqrt{2})$

Step 2.1.5

Multiply $3$ by $- 2$ .

$5 \sqrt{2} - 6 \sqrt{2}$

Step 2.2

Subtract $6 \sqrt{2}$ from $5 \sqrt{2}$ .

$- \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2}$

Decimal Form:

$- 1.41421356 \dots$