Question

If $\left(\begin{array}{cc}2& 4\\ -1& k\end{array}\right)$ is a nilpotent matrix of index $2,$ then $k$ equals to

• A. $2$
• B. $-3$
• C. $4$
• D. $-2$

Answer 1

The correct option is D $-2$

Nilpotency of matrix is $2,$ so square of given matrix will be Null matrix :

$\left(\begin{array}{cc}2& 4\\ -1& k\end{array}\right)\times \left(\begin{array}{cc}2& 4\\ -1& k\end{array}\right)=\left(\begin{array}{cc}0& 8+4k\\ -2-k& -4+k^2\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$

By comparing we can say that $k=-2$