Question

The nilpotency index of matrix $\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]$ is

Answer 1

Given $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]$

Now, $A^2=\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]$

$\Rightarrow A^2=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

$\Rightarrow A^2=0$

Hence, nilpotency index of $A$ is $2.$