If P is a 3×3 matrix such that PT=3P+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

PT=3P+I P=3PT+IT=3(3P+I)+I P=9P+4I⇒−8P=4I⇒2P=−I⇒2PX=−X

If P is a 3×3 matrix such that PT=3P+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

PT=3P+I P=3PT+IT=3(3P+I)+I P=9P+4I⇒−8P=4I⇒2P=−I⇒2PX=−X

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If P is a $3 \times 3$ matrix such that $P^{T} = 3 P + I,$ where $P^{T}$ is the transpose of P and I is the $3 \times 3$ identity matrix, then there exists a column matrix $X = [\begin{matrix} x \\ y \\ z \end{matrix}] \neq [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that

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$P X = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

PX = X

PX = –2X

2PX = –X

answer is D.

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$P^{T} = 3 P + I$
$P = 3 P^{T} + I^{T} = 3 (3 P + I) + I$
$P = 9 P + 4 I \Rightarrow - 8 P = 4 I \Rightarrow 2 P = - I \Rightarrow 2 P X = - X$

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If P is a 3×3 matrix such that PT=3P+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

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If P is a $3 \times 3$ matrix such that $P^{T} = 3 P + I,$ where $P^{T}$ is the transpose of P and I is the $3 \times 3$ identity matrix, then there exists a column matrix $X = [\begin{matrix} x \\ y \\ z \end{matrix}] \neq [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that