If $P$ is a $3 imes 3$ matrix such that $P^{T}=2 P+I$, where $P^{T}$ is the transpose of $P$ and $I$ is the $3 imes 3$ identity matrix, then there exists a column matrix $\quad X=\left[\begin{array}{l}x \ y \ z\end{array} ight]
eq\left[\begin{array}{l}0 \ 0 \ 0\end{array} ight]$ such…

Question

If $P$ is a $3 	imes 3$ matrix such that $P^{T}=2 P+I$, where $P^{T}$ is the transpose of $P$ and $I$ is the $3 	imes 3$ identity matrix, then there exists a column matrix $\quad X=\left[\begin{array}{l}x \ y \ z\end{array}ight] 
eq\left[\begin{array}{l}0 \ 0 \ 0\end{array}ight]$ such that

Aicharya · Accepted Answer

$P^{T}=2 P+I$ $\begin{array}{l} \Rightarrow P=2 P^{T}+I \Rightarrow P=2(2 P+I)+I \ \Rightarrow P=4 P+3 I \Rightarrow P+I=0 \ \Rightarrow P X+X=0 \Rightarrow P X=-X \end{array}$

If $P$ is a $3 \times 3$ matrix such that $P^{T}=2 P+I$, where $P^{T}$ is the transpose of $P$ and $I$ is…

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