Question

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ be such that $A^3=0$. Find sum of all the elements of $A^2$.

Answer 1

Given ,$A^3=0$
$\Rightarrow |A{|}^3=0$
$\Rightarrow |A|=0$
$\Rightarrow ad-bc=0$ ....(1)
Now, $A^2=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+dc& bc+d^2\end{array}\right]$
$=\left[\begin{array}{cc}{a}^2+ad& ab+bd\\ ac+dc& ad+d^2\end{array}\right]$ (by (1))
$=\left[\begin{array}{cc}a(a+d)& b(a+d)\\ (a+d)c& d(a+d)\end{array}\right]$
$=(a+d)\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$\Rightarrow A^2=(a+d)A$ .....(2)
Now, since, $A^3=O$
$A^{2}A=O$
$(a+d)AA=O$
$\Rightarrow (a+d{)}^{2}A=O$
$\Rightarrow a+d=0$ ($\because$ A is not a zero matrix)
Hence, from (2), we get
$A^2=O$