If P is a $3 \times 3$ matrix such that $\mathrm{P}^{\top}=2 \mathrm{P}+\mathrm{I}$, where $\mathrm{P}^{\top}$ is the transpose of P and I is the $3 \times 3$ identity matrix then there exists a column matrix $X=\left[\begin{array}{c}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that