Question

# If $$P$$ is a $$3\times 3$$ matrix such that $$P^{T}=2P+I$$, where $$P^{T}$$' is the transpose of $$P$$ and $$I$$ is the $$3\times 3$$ identity matrix, then there exists a column matrix $$X=\left [ \begin{array}{l}x\\y\\z\end{array}\right] \neq\left[\begin{array}{l}0\\0\\0\end{array}\right]$$ such that

A
PX=000
B
PX=X
C
PX=2X
D
PX=X
E
PX=200
F
PX=3X
G
PX=5X
H
PX=3X

Solution

## The correct option is D $$PX=-X$$Given that $$P^T=2P+I$$$$\Rightarrow (P^T)^T=(2P+I)^T$$$$\Rightarrow P=2P^T+I$$$$\Rightarrow P=2(2P+I)+I$$$$\Rightarrow 3P=-3I$$$$\Rightarrow P=-I$$$$\therefore PX=-IX$$$$=-X$$Mathematics

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