Question

If for a matrix $A,A^2+I=O$ where $I$ is the identity matrix, then $A$ equals

• A. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}i& 0\\ i& -i\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{cc}i& 0\\ i& -i\end{array}\right]$

$A^2+I=0$ I is identity matrix

$A^2=-I$

Clearly A is a matrix with complex number

From the option only option B has imaginary values verifying option B

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

$A^2=\left[\begin{array}{cc}i& 0\\ i& -i\end{array}\right]\left[\begin{array}{cc}i& 0\\ i& -i\end{array}\right]$

$\left[\begin{array}{cc}{i}^2& 0\\ i^2-i^2& i^2\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]=-I$

$\Rightarrow A^2=-I\Rightarrow A^2+I=0$

Hence option B correct

Option A represents I and option D represents -I which on squaring will give I

Hence option A and D are incorrect

Option C

$A=\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right]A^2=\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right]$

$\left[\begin{array}{cc}-1& 4\\ -2& -1\end{array}\right]\ne -I$

Hence option C is incorrect