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 = \left[\right. x \\ y \\ z \left]\right. \neq \left[\right. 0 \\ 0 \\ 0 \left]\right. such that

Engineering

Mathematics

Symmetric and Skew Symmetric Matrices

Question

If P is a 3 × 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 × 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

PX = X

PX = –X

$PX = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

PX = 2X

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P^T = 2P + I

$\Rightarrow {(P^{T})}^{T}$ = 2P^T + I^T or P = 2P^T + I

$\Rightarrow$ P = 2 (2P + I) + I or P = – I $\Rightarrow$ PX = – X.

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