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If P is a 3xx3 matrix such that P^(T) =...

If P is a `3xx3 ` matrix such that `P^(T) = 2P +I, ` where `P^(T) ` is the transpose f P and I is the `3xx3 ` identify 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`

Text Solution

Verified by Experts

The correct Answer is:
D
`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`
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