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 column matrix \[X = \left[ \begin{array}{l}
x\\
y\\
z
\end{array} \right] \ne \left[ \begin{array}{l}
0\\
0\\
0
\end{array} \right]\] such that
A.
\[PX = \left[ \begin{array}{l}
0\\
0\\
0
\end{array} \right]\]
B.
$$PX = X$$
C.
$$PX = 2X$$
D.
$$PX = - X$$
Answer :
$$PX = - X$$
Solution :
We have $${P^T} = 2P + I$$
$$\eqalign{
& \Rightarrow \,\,P = 2{P^T} + I \cr
& \Rightarrow \,\,P = 2\left( {2P + I} \right) + I \cr
& \Rightarrow \,\,P = 4P + 3I \cr
& \Rightarrow \,\,P + I = 0 \cr
& \Rightarrow \,\,PX + X = 0 \cr
& \Rightarrow \,\,PX = - X \cr} $$