If $A$is $2 \times 2$ matrix such that $A^2$ = O, then$tr \left(A\right)$ is

Let $A=\left[\begin{array}{l}a\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}b\\ c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}d\end{array}\right]$, then $A^2=\left[\begin{array}{l}{a}^2+bc\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}ab+bd\\ ac+cd\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}bc+d^2\end{array}\right]$

Now,$A^2=O$ 

$\begin{array}{r}\Rightarrow a^2+bc=0,(a+d)b=0\\ (a+d)c=0,bc+d^2=0\end{array}$ 

Suppose $a + d\ne 0,$ then $b = 0, c = 0$

$\therefore a^2=0,d^2=0\Rightarrow a=0,d=0$

Contradiction.

 Thus,$a + d = 0$ we get $tr\left(A\right) = 0$