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

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

The correct option is C 2

$A=\left[\begin{array}{cc}sin\theta & 0\\ 0& -sin\theta \end{array}\right]$

$A^T={\left[\begin{array}{cc}sin\theta & 0\\ 0& -sin\theta \end{array}\right]}^T=\left[\begin{array}{cc}sin\theta & 0\\ 0& -sin\theta \end{array}\right]$

Given $A+A^T$ is a null matrix

$\Rightarrow A+A^T=0$

$\Rightarrow \left[\begin{array}{cc}sin\theta & 0\\ 0& -sin\theta \end{array}\right]+\left[\begin{array}{cc}sin\theta & 0\\ 0& -sin\theta \end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}2sin\theta & 0\\ 0& -2sin\theta \end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow 2\sin \theta =0and-2\sin \theta =0$ on comparing both sides

$\Rightarrow \sin \theta =0$

$\therefore \theta =n\pi$

Number of values of $\theta =\{0,\pi \}$ in $[0,2\pi ]$