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 $3\times 3$ identity, 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 $PX=$

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

Answer 1

Answer: (D)

$-X$

Given $P^T=2P+I$
$(P^T{)}^T=2P^T+I$
$\Rightarrow P=2(2P+I)+I$
$\Rightarrow -3P=3I$
$\Rightarrow P=-I$
$\Rightarrow PX=-IX$
$\Rightarrow PX=-X$