If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2P + I$, 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

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

$P^{T}=2P+I$

$P:\ 3\times 3,\quad I:\ 3\times 3 \text{ identity}$

$\text{Find } X\neq 0 \text{ with } PX=\lambda X \text{ among } \lambda\in\{0,1,2,-1\}$

$\text{Transpose both sides: } (P^{T})^{T}=(2P+I)^{T}$

$\Rightarrow P=2P^{T}+I$

$P^{T}=2P+I \quad \cdots(1)$

$P=2P^{T}+I \quad \cdots(2)$

$\text{Substitute (1) in (2): } P=2(2P+I)+I$

$P=4P+2I+I=4P+3I$

$P=4P+3I \Rightarrow -3P=3I$

$\Rightarrow P=-I$

$P^{T}=(-I)^{T}=-I,\quad 2P+I=2(-I)+I=-I$

$\Rightarrow P=-I \text{ satisfies (1)}$

$PX=(-I)X=-X \quad \text{for every } X\neq 0$

$\Rightarrow \text{option } (4)$