Question

If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right],$ then AA^T = ___________.

Answer 1

The given matrix is $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right].$

$\therefore A{A}^T$

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

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

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

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


If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right],$ then AA^T = $\underline{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]}$.