If P is a `3xx3` matrix such that `P^T = 2P+I`, where `P^T` is the transpose of P and I is the `3xx3` identity matrix, then there exists a column matrix, `X = [[x],[y],[z]]!=[[0],[0],[0]]` such that

Options

A. `PX=[[0],[0],[0]]`
B. `PX = X`
C. `PX = 2X `
D. `PX =-X`

Answer

D

Explanation

`because P^(T) = 2 P +I ` ...(i)
`therefore (P^(T))^(T)=(2P+I)^(T) `
`rArr P = 2 P^(T) + I` ..(ii)
From Eqs. (i) and (ii), we get
`P = 2 (2P+I)+I`
`rArr P = -I`
`therefore PX=-IX=-X`