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

Answer: (A)

$2$

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

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

$\Rightarrow A^2=0$

$\therefore$ $A$ is nilpotent of order $2.$

Hence, option A.