Question

Let $P$ and $Q$ be $3\times 3$ matrices with $P\ne Q$. If $P^3=Q^3$ and $P^{2}Q=Q^{2}P$, then determinant of $(P^2+Q^2)$ is equal to:

• A. $-2$
• B. $1$
• C. $0$
• D. $-1$

Answer 1

The correct option is C $0$

$P^3=Q^3$

$\Rightarrow P^3-P^{2}Q=Q^3-Q^{2}P$

$\Rightarrow P^2(P-Q)=Q^2(Q-P)$

$\Rightarrow P^2(P-Q)-Q^2(Q-P)=0$

$\Rightarrow (P^2+Q^2)(P-Q)=0$ $\Rightarrow |P^2+Q^2|=0$

Hence, option 'C' is correct.