Question

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

• A. $ \begin{bmatrix}1&0\\ 0&1\end{bmatrix} $
• B. $ \begin{bmatrix}i&0\\ 0&i\end{bmatrix} $
• C. $ \begin{bmatrix}i&0\\ 0&1\end{bmatrix} $
• D. $ \begin{bmatrix}-1&0\\ 0&-1\end{bmatrix} $

Answer 1

Answer: (B)

Let $A =\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} $
$ \therefore A^{2} =\begin{bmatrix} i & 0 \\ 0 & i\end{bmatrix}\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} $
$=\begin{bmatrix} -1 & 0 \\ 0 & -1\end{bmatrix} $
$ \therefore A^{2}+I =\begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}+\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} $
$=\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}=0$, which is true.