Question

$A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]$, then A is a nilpotent matrix of index

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is A $2$

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

For $A$ to be nilpotent matrix then $A^m=0$

where m is the index, so let's check for m=2

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

$A=\left[\begin{array}{ccc}1+2-3& 2+4-6& 3+6-9\\ 1+2-3& 2+4-6& 3+6-9\\ -1-2+3& -2-4+6& -3-6+9\end{array}\right]$

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

$\therefore$ For $m=2$ ,$A$ is a Nilpotent matrix