Linear Algebra Examples

$[\begin{array}{c} \cos (x) & - \sin (x) \\ \sin (x) & \cos (x) \end{array}]$

Step 1

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Step 2

Find the determinant.

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Step 2.1

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

$\cos (x) \cos (x) - \sin (x) (- \sin (x))$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $\cos (x) \cos (x)$ .

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Step 2.2.1.1.1

Raise $\cos (x)$ to the power of $1$ .

$\cos^{1} (x) \cos (x) - \sin (x) (- \sin (x))$

Step 2.2.1.1.2

Raise $\cos (x)$ to the power of $1$ .

$\cos^{1} (x) \cos^{1} (x) - \sin (x) (- \sin (x))$

Step 2.2.1.1.3

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

${\cos (x)}^{1 + 1} - \sin (x) (- \sin (x))$

Step 2.2.1.1.4

Add $1$ and $1$ .

$\cos^{2} (x) - \sin (x) (- \sin (x))$

Step 2.2.1.2

Multiply $- \sin (x) (- \sin (x))$ .

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Step 2.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\cos^{2} (x) + 1 \sin (x) \sin (x)$

Step 2.2.1.2.2

Multiply $\sin (x)$ by $1$ .

$\cos^{2} (x) + \sin (x) \sin (x)$

Step 2.2.1.2.3

Raise $\sin (x)$ to the power of $1$ .

$\cos^{2} (x) + \sin^{1} (x) \sin (x)$

Step 2.2.1.2.4

Raise $\sin (x)$ to the power of $1$ .

$\cos^{2} (x) + \sin^{1} (x) \sin^{1} (x)$

Step 2.2.1.2.5

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

$\cos^{2} (x) + {\sin (x)}^{1 + 1}$

Step 2.2.1.2.6

Add $1$ and $1$ .

$\cos^{2} (x) + \sin^{2} (x)$

Step 2.2.2

Rearrange terms.

$\sin^{2} (x) + \cos^{2} (x)$

Step 2.2.3

Apply pythagorean identity.

$1$

Step 3

Since the determinant is non-zero, the inverse exists.

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{1} [\begin{array}{c} \cos (x) & \sin (x) \\ - \sin (x) & \cos (x) \end{array}]$

Step 5

Divide $1$ by $1$ .

$1 [\begin{array}{c} \cos (x) & \sin (x) \\ - \sin (x) & \cos (x) \end{array}]$

Step 6

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 1 \cos (x) & 1 \sin (x) \\ 1 (- \sin (x)) & 1 \cos (x) \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $\cos (x)$ by $1$ .

$[\begin{array}{c} \cos (x) & 1 \sin (x) \\ 1 (- \sin (x)) & 1 \cos (x) \end{array}]$

Step 7.2

Multiply $\sin (x)$ by $1$ .

$[\begin{array}{c} \cos (x) & \sin (x) \\ 1 (- \sin (x)) & 1 \cos (x) \end{array}]$

Step 7.3

Multiply $- \sin (x)$ by $1$ .

$[\begin{array}{c} \cos (x) & \sin (x) \\ - \sin (x) & 1 \cos (x) \end{array}]$

Step 7.4

Multiply $\cos (x)$ by $1$ .

$[\begin{array}{c} \cos (x) & \sin (x) \\ - \sin (x) & \cos (x) \end{array}]$