Question

If P is $3\times 3$ 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 $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\ne \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$ such that

• A. $PX=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
• B. PX = X
• C. PX = 2X
• D. PX = -X

Answer 1

The correct option is D PX = -X

Given $P^T=2P+I$
Taken transpose
$\Rightarrow P=2P^T+I$
$\Rightarrow P=2(2P+I)+1\Rightarrow P+I=0\Rightarrow PX+X=0\therefore PX=-X$