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

$\begin{bmatrix}0\\0\\0\end{bmatrix}$ $X$ $2X$ $-X$

$-X$

We have, $P^T = 2P + I$ $⇒(P^T)^T =(2P+I)^T$ $⇒P = 2P^T + I$ $⇒P=2(2P+I) + I$ $⇒P = 4P + 3I$ $⇒3P+3I=O$ $⇒P+I=O$ $⇒P=-I$ $⇒PX=-IX$ $⇒PX=-X$