Question

If $A=\left[\begin{array}{cc}2& 4\\ -1& k\end{array}\right]$ and $A^2=O$ (null matrix), then find the value of $k$.

Answer 1

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

Therefore, $A^2=\left[\begin{array}{cc}2& 4\\ -1& k\end{array}\right]\times \left[\begin{array}{cc}2& 4\\ -1& k\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}4-4& 8+4k\\ -2-k& -4+k^2\end{array}\right]=0$

$\Rightarrow \left[\begin{array}{cc}0& 8+4k\\ -2-k& -4+k^2\end{array}\right]=0$

$\Rightarrow 8+4k=0$

$\Rightarrow 8=-4k$

$\Rightarrow k=-2$