Question

Let $A=\left[\begin{array}{cc}\sin \theta & 0\\ 0& -\sin \theta \end{array}\right]$. If $A+A^T$ is a null matrix, then the number of values of $\theta$ in $[0,2\pi ]$ is

Answer 1

$A=\left[\begin{array}{cc}\sin \theta & 0\\ 0& -\sin \theta \end{array}\right]A^T={\left[\begin{array}{cc}\sin \theta & 0\\ 0& -\sin \theta \end{array}\right]}^T=\left[\begin{array}{cc}\sin \theta & 0\\ 0& -\sin \theta \end{array}\right]$
Given $A+A^T$ is a null matrix
$\Rightarrow A+A^T=0\Rightarrow \left[\begin{array}{cc}\sin \theta & 0\\ 0& -\sin \theta \end{array}\right]+\left[\begin{array}{cc}\sin \theta & 0\\ 0& -\sin \theta \end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow \left[\begin{array}{cc}2\sin \theta & 0\\ 0& -2\sin \theta \end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
on comparing both sides
$\Rightarrow 2\sin \theta =0$
$\Rightarrow \sin \theta =0\therefore \theta =n\pi$
Values of $\theta$ are $\{0,\pi ,2\pi \}$ in $[0,2\pi ]$