Question

Find the adjoint matrix and inverse matrix of the matrix
$\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$

Answer 1

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

Now, determinant of matrix $A$ will be,

$|A|=\left|\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right|={\cos }^2\alpha +{\sin }^2\alpha =1$

Since, $|A|\ne 0$, inverse of $A$ exists.

Now, adj $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$

Therefore, $A^{-1}=\frac{1}{|A|}$adj $A$

$\Rightarrow A^{-1}=1\times \left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$

Hence, $A^{-1}=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$