Question

If the matrix $\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$ is singular, then $\theta =$

• A. $\pi$
• B. $\pi /2$
• C. $\pi /3$
• D. $\pi /4$

Answer 1

The correct option is D $\pi /4$

Let, $A=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$
We know that, a square matrix A is called singular if, $\text{det(A)}=0$
$=>\left|\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right|=0$
Expanding along third row,
${\cos }^2\theta -{\sin }^2\theta =0$

$\Rightarrow {\tan }^2\theta =1$

$\Rightarrow \theta =\pi /4$