Question

If for a matrix $A,A^2+I=0$ , where $I$ is the Identity matrix of order $2$, then $A$ =

• A. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$
• C. $\left[\begin{array}{cc}i& 0\\ 0& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Answer 1

Answer: (B)

Let $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$
$\therefore A^2=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$
$=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
$\therefore A^2+I=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=0$, which is true.