First slide

Applications of determinant

Question

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

Moderate

A

$A$
B

$I$
C

$A^{1000}$
D

none of these

Solution

$\begin{array}{l} Q = P A P^{T} \\ ∴ x = P^{T} Q^{1000} P \\ x = P^{T} {(P A P^{T})}^{1000} P \\ x = P^{T} P A P^{T} {(P A P^{T})}^{999} P \\ x = I A P^{T} P A P^{T} {(P A P^{T})}^{998} P \\ x = A I A P^{T} {(P A P^{T})}^{998} P \\ x = A^{2} P^{T} P A P^{T} {(P A P^{T})}^{997} P \end{array}$
$\begin{array}{l} x = A^{3} P^{T} {(P A P^{T})}^{997} P \\ ⋮ \\ x = A^{1000} (∵ A is involutary) \\ x = I \end{array}$

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