Question

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

• A. $P X=\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}$
• B. $PX =X$
• C. $PX = 2X$
• D. $PX = -X$

Answer 1

Answer: (D)

Given that $P^{T}=2 P+1$.
$\Rightarrow P=2 P^{T}+1=2(2 P+1)+1$
$\Rightarrow P+I=0 $
$\Rightarrow P X+X=0$
Therefore, $P X=-X$