Question

$P$ is an orthogonal matrix and $A$ is a periodic matrix with period $4$ and $Q=PA{P}^T$ then $X=P^T{Q}^{2021}P$

• A. $A$
• B. $A^2$
• C. $A^3$
• D. $A^4$

Answer 1

The correct option is A $A$

$\begin{array}{c}Q=PA{P}^T\\ X=P^T{Q}^{2021}P\\ =P^T{\left(PA{P}^T\right)}^{2021}P\\ =P^T{\left(PA{P}^T\right)}^{2020}PA{P}^{T}P\\ =P^T{\left(PA{P}^T\right)}^{2020}PA\\ P^{T}P=1\\ =P^T\left(PA{P}^T\right)2019PA{P}^{T}PA\\ =P^T{\left(PA{P}^T\right)}^{2019}P{A}^2\\ ={\left(A\right)}^{2021}\\ ={\left({\left(A\right)}^4\right)}^{505}A\\ =A\end{array}$

Hence, this is the answer.