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If P is an orthogonal matrix and Q=P A P...

If `P` is an orthogonal matrix and `Q=P A P^T an dx=P^T` `A` b. `I` c. `A^(1000)` d. none of these

A

`A`

B

`I`

C

`A^(1000)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`P^(T)P=I`
`Q=PAP^(T)`
`:. X=P^(T) Q^(1000)P=P^(T) (PAP^(T))^(1000) P`
`=P^(T)PAP^(T) (PAP^(T))^(999)P`
`=IAP^(T). PAP^(T) (PAP^(T))^(998) P`
`=AIAP^(T) (PAP^(T))^(998)P`
`=A^(2)P^(T) PAP^(T) (PAP^(T))^(997)P`
`=A^(3) P^(T) (PAP^(T))^(997)P`

`=A^(1000)=I" "( :' " A is involuntary")`
Hence, `x^(-1)=I`.
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