Discussion Forum

If P is a 3x3 matrix such that $P^{T} =2P+I$, where $p^{T}$  is the transpose of P and I is the 3 x 3 identity matrix, then there exists a column matrix

$X=\begin{bmatrix}x \\y \\z \end{bmatrix}\neq\begin{bmatrix}0 \\0 \\0 \end{bmatrix}$ such that

Answer:

Explanation:

 Given, $P^{T}=2P+I$ ...........(i)

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

 $\Rightarrow$    $P=2P^{T}+I$

$\Rightarrow$   $P=2(2P+I)+I$

$\Rightarrow$   $P=4P+3I$ or $3P=-3I$

$\Rightarrow$  PX=-IX