If P is a 3 × 3 matrrix such that P^T = 2P + I where P^T is the transpose of P and I is 3 × 3 identity matrix then there exists a column matrix $X = [\begin{array}{l} x \\ y \\ z \end{array}] \neq [\begin{array}{l} 0 \\ 0 \\ 0 \end{array}]$ such that

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

PX = X

PX = 2X

PX = –X

answer is D.

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Detailed Solution

we have $P^{T} = 2 P + I \Rightarrow P^{T} - 2 P = I - (1) \Rightarrow P - 2 P^{T} = I - (2)$
$\Rightarrow P = - I \Rightarrow (P + I) X = 0 \Rightarrow P X = - X$

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