Question

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

• A. Involutory matrix
• B. Idempotent matrix
• C. Nilpotent matrix
• D. none of these

Answer 1

Answer: (C)

Nilpotent matrix

$A^2=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$A^2=0$ $\Rightarrow$ $A\to$ Nilpotent
Ans: C