Question

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

• A. $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
• B. $\left[\begin{array}{cc}\sin \theta & \cos \theta \\ \cos \theta & \sin \theta \end{array}\right]$
• C. $\left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]$
• D. $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$

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

$|A|={\cos }^2\theta +{\sin }^2\theta =1$

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

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

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