If $P$ and $Q$ are two $3 imes 3$ matrices such that $|P Q|=1$ and $|P|=9$, then the determinant of adjoint the matrix $P$. Adj $3 Q$ is

Question

If $P$ and $Q$ are two $3 	imes 3$ matrices such that $|P Q|=1$ and $|P|=9$, then the determinant of adjoint the matrix $P$. Adj $3 Q$ is

Aicharya · Accepted Answer

$\begin{aligned} & |P||Q|=1 \Rightarrow|Q|=\frac{1}{9} \ & \Rightarrow \mid \operatorname{adj}(P \operatorname{adj}(3 Q)|=|\operatorname{adj} \operatorname{adj}(3 Q)|| \operatorname{adj} P \mid \ & =|3 Q|^4|P|^2=\left(3^3ight)^4 	imes|Q|^4|P|^2 \ & =3^{12} 	imes \frac{1}{9^4} 	imes 9^2=\frac{3^{12}}{3^4}=3^8=9^4 .\end{aligned}$

If $P$ and $Q$ are two $3 \times 3$ matrices such that $|P Q|=1$ and $|P|=9$, then the determinant of…

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