Linear Algebra Examples

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

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

Step 1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

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

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

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

$5$ to the left of

\sqrt{2}

$\sqrt{2}$ .

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

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

Step 2.1.2

Multiply

- \sqrt{6} (2 \sqrt{3})

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

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Step 2.1.2.1

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 2.1.2.2

Combine using the product rule for radicals.

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

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

Step 2.1.2.3

Multiply

3

$3$ by

6

$6$ .

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

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

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

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

Step 2.1.3

Rewrite

18

$18$ as

3^{2} \cdot 2

$3^{2} \cdot 2$ .

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Step 2.1.3.1

Factor

9

$9$ out of

18

$18$ .

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

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

Step 2.1.3.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

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

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

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

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

Step 2.1.5

Multiply

3

$3$ by

- 2

$- 2$ .

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

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

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

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

Step 2.2

Subtract

6 \sqrt{2}

$6 \sqrt{2}$ from

5 \sqrt{2}

$5 \sqrt{2}$ .

- \sqrt{2}

$- \sqrt{2}$

- \sqrt{2}

$- \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

- \sqrt{2}

$- \sqrt{2}$

Decimal Form:

- 1.41421356 \dots

$- 1.41421356 \dots$