Mathematics - Matrix Question with Solution | TestHub

$A^3=B^3\dots (i)$

$$\begin{array}{rl}{A}^{2}B& =B^{2}A\\ A^{2}B-B^{2}A& =0\\ AB(A-B)& =0\end{array}$$

Since $A\ne B$, we have $A-B\ne 0$.

Therefore, $AB=0$.

If $A=0$, then from equation (i), $B^3=0$, which implies $B=0$.

If $B=0$, then from equation (i), $A^3=0$, which implies $A=0$.

Thus, $A=0$ and $B=0$.

Now consider the expression $(A^2+B^2)(A-B)=0$.

Since $|A-B|\ne 0$, it follows that $A-B\ne 0$.

Therefore, $A^2+B^2=0$.

Since $A$ and $B$ are real numbers, $A^2\ge 0$ and $B^2\ge 0$.

The only way for $A^2+B^2=0$ is if $A^2=0$ and $B^2=0$, which means $A=0$ and $B=0$..(i)