Matrices and Determinants 3 Question 7
9. 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 the $3 \times 3$ identity matrix, then $x \quad 0$ there exists a column matrix, $X=$
$\begin{bmatrix} x \\ y\\ z \end{bmatrix}$ $\neq$ $\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}$ such that
(a) $PX=$ $\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}$
(b) $P X=X$
(c) $P X=2 X$
(d) $P X=-X$
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Answer:
Correct Answer: 9. (d)
Solution:
- Given, $P^{T}=2 P+I$
$ \begin{array}{ll} \therefore & \left(P^{T}\right)^{T}=(2 P+I)^{T}=2 P^{T}+I \\ \Rightarrow & P=2 P^{T}+I \\ \Rightarrow & P=2(2 P+I)+I \\ \Rightarrow & P=4 P+3 I \text { or } 3 P=-3 I \\ \Rightarrow & P X=-I X=-X \end{array} $
