Question

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, $X =\begin {bmatrix}x \\ y \\z \end {bmatrix} \ne \begin{bmatrix} 0\\0\\0 \end{bmatrix}$ such that

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

Answer 1

Answer: (B)

Given, $P^T =2 P+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=-3P \Rightarrow PX=-IX=X$