1) If P is a 3x3 matrix such that $P^{T} =2P+I$, where $p^{T}$ is the transpose of P and I is the 3 x 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) $PX=\begin{bmatrix}0 \\0 \\0 \end{bmatrix}$ B) PX=X C) PX=2X D) PX=-X Answer: Option DExplanation: Given, $P^{T}=2P+I$ ...........(i) $\therefore$ $(P^{T})^{T}=(2P+I)^{T}=2P^{T}+I$ $\Rightarrow$ $P=2P^{T}+I$ $\Rightarrow$ $P=2(2P+I)+I$ $\Rightarrow$ $P=4P+3I$ or $3P=-3I$ $\Rightarrow$ PX=-IX
