Find the inverse of each of the following matrices: (i) $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$ (ii) $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$

Find the inverse of each of the following matrices:
(i) $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$
(ii) $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$

Solution:

(i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero.

Now, $|A|=\cos \theta(\cos \theta)+\sin \theta(\sin \theta)$

$=1$

Therefore, $A^{-1}$ exists.

Cofactors of $A$ are

$C_{11}=\cos \theta$

$C_{12}=\sin \theta$

$C_{21}=-\sin \theta$

$C_{22}=\cos \theta$

As, $\operatorname{adj} A=\left[\begin{array}{ll}C_{11} & C_{12} \\ C_{21} & C_{22}\end{array}\right]^{T}$

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

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

Now, $A^{-1}=\frac{1}{|A|} \cdot \operatorname{adj} A$

$\begin{aligned} &A^{-1}=\frac{1}{1}\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] \\ &A^{-1}=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] \end{aligned}$