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If $P$ is a $3 \times 3$ matrix such that $P^{T}=2 P+I$, where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $\quad X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that
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Verified Answer
The correct answer is:
$P X=-X$
$P^{T}=2 P+I$
$\begin{array}{l}
\Rightarrow P=2 P^{T}+I \Rightarrow P=2(2 P+I)+I \\
\Rightarrow P=4 P+3 I \Rightarrow P+I=0 \\
\Rightarrow P X+X=0 \Rightarrow P X=-X
\end{array}$
$\begin{array}{l}
\Rightarrow P=2 P^{T}+I \Rightarrow P=2(2 P+I)+I \\
\Rightarrow P=4 P+3 I \Rightarrow P+I=0 \\
\Rightarrow P X+X=0 \Rightarrow P X=-X
\end{array}$
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