Linear Algebra Examples

$[\begin{array}{c} \cos (0) & - \sin (0) \\ \sin (0) & \cos (0) \end{array}]$

Step 1

The exact value of $\cos (0)$ is $1$ .

$[\begin{array}{c} 1 & - \sin (0) \\ \sin (0) & \cos (0) \end{array}]$

Step 2

The exact value of $\sin (0)$ is $0$ .

$[\begin{array}{c} 1 & - 0 \\ \sin (0) & \cos (0) \end{array}]$

Step 3

Multiply $- 1$ by $0$ .

$[\begin{array}{c} 1 & 0 \\ \sin (0) & \cos (0) \end{array}]$

Step 4

The exact value of $\sin (0)$ is $0$ .

$[\begin{array}{c} 1 & 0 \\ 0 & \cos (0) \end{array}]$

Step 5

The exact value of $\cos (0)$ is $1$ .

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 6

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 7

Find the determinant.

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Step 7.1

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

$1 \cdot 1 + 0 \cdot 0$

Step 7.2

Simplify the determinant.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $1$ by $1$ .

$1 + 0 \cdot 0$

Step 7.2.1.2

Multiply $0$ by $0$ .

$1 + 0$

Step 7.2.2

Add $1$ and $0$ .

$1$

Step 8

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

Step 9

Substitute the known values into the formula for the inverse.

$\frac{1}{1} [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 10

Divide $1$ by $1$ .

$1 [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 11

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 1 \cdot 1 & 1 \cdot 0 \\ 1 \cdot 0 & 1 \cdot 1 \end{array}]$

Step 12

Simplify each element in the matrix.

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Step 12.1

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 \cdot 0 \\ 1 \cdot 0 & 1 \cdot 1 \end{array}]$

Step 12.2

Multiply $0$ by $1$ .

$[\begin{array}{c} 1 & 0 \\ 1 \cdot 0 & 1 \cdot 1 \end{array}]$

Step 12.3

Multiply $0$ by $1$ .

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \cdot 1 \end{array}]$

Step 12.4

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$