Let P and Q be [3 3] matrices [P Q.] If P^3 = Q^3 and P^2Q = Q^2P then determinant of [( P…

Question

Let $P$ and $Q$ be $$3 \times 3$$ matrices $$P \ne Q.$$ If ${P^3} = {Q^3}$ and

${P^2}Q = {Q^2}P$ then determinant of $$\left( {{P^2} + {Q^2}} \right)$$ is equal to :

Accepted Answer

Given

$${P^3} = {q^3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

$${P^2}Q = {Q^2}p\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$

Subtracting $(1)$ and $(2)$, we get

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

$$ \Rightarrow {P^2}\left( {P - Q} \right) + {Q^2}\left( {P - Q} \right) = 0$$

$$ \Rightarrow \left( {{P^2} + {Q^2}} \right)\left( {P - Q} \right) = 0$$

$$ \Rightarrow \left| {{p^2} + {Q^2}} \right| = 0$$

as $$P \ne Q$$

Solutions · 1