Question

If $A=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]$, then $A^2$ is equal

• A. $O$
• B. $I$
• C. $-I$
• D. none of these

Answer 1

The correct option is A $O$

$A=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]$
$A^2=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]$
$=\left[\begin{array}{cc}{a}^2{b}^2-a^2{b}^2& a{b}^3-a{b}^3\\ -a^{3}b+a^{3}b& -a^2{b}^2+a^2{b}^2\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\therefore A^2=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=O$
Hence, option A.