Question

If $A=\left[\begin{array}{cc}2& 4\\ -1& k\end{array}\right]$ and $A^2=0$, find $k$

Answer 1

$\begin{array}{c}{A}^2=0\\ \left[\begin{array}{cc}2& 4\\ -1& k\end{array}\right]\left[\begin{array}{cc}2& 4\\ -1& k\end{array}\right]=0\\ \Rightarrow \left[\begin{array}{cc}0& 8-4k\\ -2-k& -4+k^2\end{array}\right]=0\\ \Rightarrow \left(-2-k\right)\left(8-4k\right)=0\\ -2-k=0or8-4k=0\\ \therefore k=-2or4k=8,k=2\end{array}$