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Let $P$ and $Q$ be $3 \times 3$ matrices with $P \neq Q$. If $P^3=Q^3$ and $P^2 Q=Q^2 P$, then determinant of $\left(P^2+Q^2\right)$ is equal to
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The correct answer is:
$0$
$0$
$P^3=Q^3$
$P^3-P^2 Q=Q^3-Q^2 P$
$P^2(P-Q)=Q^2(Q-P)$
$P^2(P-Q)+Q^2(P-Q)=O$
$\left(P^2+Q^2\right)(P-Q)=O \quad \Rightarrow\left|P^2+Q^2\right|=0$
$P^3-P^2 Q=Q^3-Q^2 P$
$P^2(P-Q)=Q^2(Q-P)$
$P^2(P-Q)+Q^2(P-Q)=O$
$\left(P^2+Q^2\right)(P-Q)=O \quad \Rightarrow\left|P^2+Q^2\right|=0$
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