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

• A. $PX=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
• B. $PX=X$
• C. $PX=2X$
• D. $PX=-X$
• E. $PX=\left[\begin{array}{c}2\\ 0\\ 0\end{array}\right]$
• F. $PX=3X$
• G. $PX=5X$
• H. $PX=-3X$

Answer 1

The correct option is D $PX=-X$

Given that $P^T=2P+I$

$\Rightarrow (P^T{)}^T=(2P+I{)}^T$

$\Rightarrow P=2P^T+I$

$\Rightarrow P=2(2P+I)+I$

$\Rightarrow 3P=-3I$

$\Rightarrow P=-I$

$\therefore PX=-IX$

$=-X$