Let P and Q be 3×3 matrices P ne Q. I | vedclass

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Question

$(i)$ Two matrices $P$ and $Q$ of order $3 	imes 3$ such that

$P 
eq Q.$

$(ii)$ $P^{3}=Q^{3}$ and $P^{2} Q=Q^{2} P$

To find The value of de

Vedclass · Accepted Answer

$0$

Vedclass · Answer

$(i)$ Two matrices $P$ and $Q$ of order $3 	imes 3$ such that

$P 
eq Q.$

$(ii)$ $P^{3}=Q^{3}$ and $P^{2} Q=Q^{2} P$

To find The value of determinant of $P^{2}+Q^{2}.$

On subtracting the given equations, we get

${P^{3}-P^{2} Q=Q^{3}-Q^{2} P}$

$\Rightarrow$ ${P^{2}(P-Q)=Q^{2}(Q-P)}$

$\Rightarrow$ $(P-Q)\left(P^{2}+Q^{2}ight)=0$

Now, since $P 
eq Q \quad[	ext { given }]$

$\Rightarrow \quad P-Q 
eq 0 \Rightarrow\left|P^{2}+Q^{2}ight|=0$

$	herefore $ $P^{2}+Q^{2}=0$

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

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