# Solve this

Question:

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

Solution:

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

$\therefore A A^{T}$

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

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

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

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

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