Q. If P is an orthogonal matrix and $\mathcal{Q =}PAP^{T}$ and $x = P^{T}\mathcal{Q}^{1000}P$, Then $x^{- 1}$ is, where $A$ is involutary matrix
A) $A$
B) $I$
C) $A^{1000}$
D) None of these
Explanation
(b)
$P^{T}P = I$
$\mathcal{Q =}PAP^{T}$
$\therefore x = P^{T}\mathcal{Q}^{1000}P = P^{T}\left( PAP^{T} \right)^{1000}P$
$= P^{T}PAP^{T}\left( PAP^{T} \right)^{999}P$
$= IAP^{T} \bullet PAP^{T}\left( PAP^{T} \right)^{998}P\$
$= AIP^{T}\left( PAP^{T} \right)^{997}P$
$= A^{2}P^{T}PAP^{T}\left( PAP^{T} \right)^{997}P$
$= A^{3}P^{T}(\left( PAP^{T} \right)^{997}P$
$\vdots$
$= A^{1000} = I\ \ \ \ (\because A\$is involuntary$)$
Hence, $x^{- 1} = I$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.