IIT-JEE 2012 · Paper 2, Mathematics question 44

If $P$ is a $3\times3$ matrix such that $P^T=2P+I$, where $P^T$ is the transpose of $P$ and $I$ is the $3\times3$ identity matrix, then there exists a column matrix $X=\begin{bmatrix}x\\y\\z\end{bmatrix}\ne\begin{bmatrix}0\\0\\0\end{bmatrix}$ such that