Question

If $P$ is an orthogonal matrix and $Q=PA{P}^T$ and $B=P^T{Q}^{1000}P$, then $B^{-1}$ is, where $A$ is involutory matrix,

• A. $A$
• B. $A^{1000}$
• C. $I$
• D. None of these

Answer 1

Answer: (C)

$I$

Given $\theta =PA{P}^T,B=P^T{Q}^{1000}P$ and $A$ is inventory matrix is $A^2=I$

$B=P^T{Q}^{1000}P=P^T(PA{P}^T{)}^{1000}P$

$=P^T\left(PA{P}^T\right)(PA{P}^T{)}^{999}P$

$=IA{P}^T\left(PA{P}^T{)}^{999}P\right[\because P^{T}P=I]$

$=A{P}^T\left(PA{P}^T\right)(PA{P}^T{)}^{998}P$

$=AIA{P}^T(PA{P}^T{)}^{998}P$

$=A^2{P}^T(PA{P}^T{)}^{998}P$

$=A^{1000}{P}^{T}P=A^{1000}=I[\because A^2=I]$

As $A=I,A^{-1}=I$

Option $C$ is correct