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 =[x y z ] ≠ [ 0 0 0 ] such that

Question

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 =[x y z ] ≠ [ 0 0 0 ] such that (A) PX=[ 0 0 0 ] (B) PX= X (C) PX=2X (D) PX=-X

Tardigrade Admin · Accepted Answer

Given, PT =2 P+I ....(i) ∴ (PT)T=(2P+I)T=2PT+I ⇒ P+2PT+I ⇒ P=2(2P+I)+I ⇒ P=4P+3I or 3P=-3P ⇒ PX=-IX=X

Q. 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, $X = ⎣ ⎡ x y z ⎦ ⎤ \neq = ⎣ ⎡ 000 ⎦ ⎤$ such that

$PX = ⎣ ⎡ 000 ⎦ ⎤$

$PX = X$

$PX = 2 X$

$PX = - X$

Given, $P^{T} = 2 P + I$ ....(i)
$∴ (P^{T})^{T} = (2 P + I)^{T} = 2 P^{T} + I$
$\Rightarrow P + 2 P^{T} + I$
$\Rightarrow P = 2 (2 P + I) + I$
$\Rightarrow P = 4 P + 3 I$
or $3 P = - 3 P \Rightarrow PX = - I X = X$

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