Question

If $A=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]$ and $A^n=O,$ then the minimum value of ${}^{\prime }{n}^{\prime }$ is

Answer 1

$A^2=A\cdot A\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]=\left[\begin{array}{cc}{a}^2{b}^2-a^2{b}^2& a{b}^3-a{b}^3\\ -a^{3}b+a^{3}b& -a^2{b}^2+a^2{b}^2\end{array}\right]=O$
$\Rightarrow A^3=A.A^2=O$ and $A^n=O,$
for all $n\ge 2$