Question

Simplify:
$\cos \theta \left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right].$

Answer 1

$cos\theta \left[\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right]+sin\theta \left[\begin{array}{cc}sin\theta & -cos\theta \\ cos\theta & sin\theta \end{array}\right]$

A matrix when multiplied by a constant, gets multiplied to each term of that matrix.

$\therefore \left[\begin{array}{cc}co{s}^2\theta & cos\theta sin\theta \\ -sin\theta cos\theta & co{s}^2\theta \end{array}\right]+\left[\begin{array}{cc}si{n}^2\theta & -sin\theta cos\theta \\ cos\theta sin\theta & si{n}^2\theta \end{array}\right]$

$=\left[\begin{array}{cc}co{s}^2\theta +si{n}^2\theta & cos\theta sin\theta -cos\theta sin\theta \\ -cos\theta sin\theta +cos\theta sin\theta & co{s}^2\theta +si{n}^2\theta \end{array}\right]$

$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$