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