P is an orthogonal matrix and A is a periodic matrix with period 4, Q=PAP^T then X=P^TQ²⁰²¹P is
equal to

We have $$\begin{gathered} {\mathrm{PP}}^{\mathrm{T}}=\mathrm{I},{\mathrm{A}}^5=\mathrm{A}\Rightarrow {\mathrm{Q}}^{2021}={\left({\mathrm{PAP}}^{\mathrm{T}}\right)}^{2021} \\ {Q}^{2021}=\mathrm{PA}{P}^{\mathrm{T}}{\mathrm{PAP}}^{\mathrm{T}}{\mathrm{PAP}}^{\mathrm{T}}........{\mathrm{PAP}}^{\mathrm{T}} \\ {Q}^{2021}={\mathrm{PA}}^{2021}{P}^{\mathrm{T}} \\ {P}^T{Q}^{2021}P=P^T{\mathrm{PA}}^{2021}{P}^{\mathrm{T}}P=A^{2021} \\ period is 4 \end{gathered}$$
$\therefore \mathrm{X}={\mathrm{A}}^{2021}=\mathrm{A}$