Question

Let $A=\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]$ such that $A^3=0$ , then sum of all the elements of $A^2$ is........

Answer 1

$A=\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]A^3=0\Rightarrow A^2\cdot A=0$

Let$A^2=0\therefore \left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]=0\therefore \left[\begin{array}{cc}{\alpha }^2+\beta \gamma & \alpha \beta +\beta \delta \\ \gamma \alpha +\delta \gamma & \gamma \beta +{\delta }^2\end{array}\right]=0\therefore {\alpha }^2+\beta \gamma =0,\alpha \beta +\beta \delta =0,\gamma \alpha +\delta \gamma =0,\gamma \beta +{\delta }^2=0$

$\therefore$ Sum of all elements is also $0$