If P is a 3 × 3 matrix such that P T=2 P+I, where P T is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix X=[x y z] ≠ [0 0 0] such that

Question

If P is a 3 × 3 matrix such that P T=2 P+I, where P T is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix X=[x y z] ≠ [0 0 0] such that (A) P X=[0 0 0] (B) PX =X (C) PX = 2X (D) PX = -X

Tardigrade Admin · Accepted Answer

Given that PT=2 P+1. ⇒ P=2 PT+1=2(2 P+1)+1 ⇒ P+I=0 ⇒ P X+X=0 Therefore, P X=-X

Q. If $P$ is a $3 \times 3$ matrix such that $PT = 2 P + I$ , where $PT$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = ⎣ ⎡ x y z ⎦ ⎤ \neq = ⎣ ⎡ 000 ⎦ ⎤$ such that

$PX = ⎣ ⎡ 000 ⎦ ⎤$

$PX = X$

$PX = 2 X$

$PX = - X$

Given that $P^{T} = 2 P + 1$ .
$\Rightarrow P = 2 P^{T} + 1 = 2 (2 P + 1) + 1$
$\Rightarrow P + I = 0$
$\Rightarrow PX + X = 0$
Therefore, $PX = - X$

Q. If P is a 3×3 matrix such that PT=2P+I, where PT is the transpose of P and I is the 3×3 identity matrix, then there exists a column matrix X=⎣⎡​xyz​⎦⎤​=⎣⎡​000​⎦⎤​ such that

PX=⎣⎡​000​⎦⎤​

PX=X

PX=2X

PX=−X