Matrices And Determinants Ques 84
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 there exists a column matrix $X=$
$\begin{bmatrix}
x \
y\`
z
$\begin{bmatrix}$
$\neq$ $\begin{bmatrix}
0
0
0
$\begin{bmatrix}$
such that the condition is met
(a) $PX=$ $\begin{bmatrix} 0 0\\ 0 $\begin{bmatrix}$
(b) $P X=X$
(c) $P X=2 X$
(d) $P X=-X$
Show Answer
Answer:
Correct Answer: 84.(d)
Solution:
Formula:
Evaluation of the Determinant:
- Given, $P^{T}=2 P+I$
$ \begin{array}{ll} \therefore & \left(P^{T}\right)^{T}=2 P^{T}+I \\ \Rightarrow & P=2 P^{T}+I \\ \Rightarrow & P=2(2L+I)+I \\ \Rightarrow & P=4P+3I \text { or } 3P=-3I \\ ⇒ & P X=I X=X \end{array} $
BETA
