Solve Matrix

$$\left. \begin{array} { l } { \text { If } A = \begin{bmatrix} \begin{array} { c c } { \cos \alpha } & { - \sin \alpha } \\ { \sin \alpha } & { \cos \alpha } \end{array} \end{bmatrix} \text { and } } \\ { B = \begin{bmatrix} \begin{array} { l r } { \cos \beta } & { - \sin \beta } \\ { \sin \beta } & { \cos \beta } \end{array} \end{bmatrix} , \text { show that } A B = B A } \end{array} \right.$$

Solve for A, B

$$A\in \mathrm{R}$$

$$B\in \mathrm{R}$$

Solution Steps

Subtract BA from both sides.

$$AB-BA=0$$

Combine AB and -BA to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any A.

$$A\in \mathrm{R}$$