# If $P$ is a $3\times 3$ matrix such that $P^T=2P+1$ where $P^T$ is the transpose of $p$ and $I$ is the $3\times 3$ identity matrix then there exists a column matrix $x=\begin{bmatrix}x\\y\\z\end{bmatrix}\neq \begin{bmatrix}0\\0\\0\end{bmatrix}$ such that

$(a)\;PX=\begin{bmatrix}0\\0\\0\end{bmatrix}\qquad(b)\;PX=x\qquad(c)PX=2x\qquad(d)\;PX=-x$

We have $P^T=2P+1$
$\Rightarrow P=2P^T+1$
$\Rightarrow P=2(2P+I)+I$
$\Rightarrow P=4P+3I$
$\Rightarrow P+I=0$
$\Rightarrow PX+x=0$
$\Rightarrow PX=-x$
Hence (d) is the correct answer.