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Inverse of A Q^n A^T where Q = A^T B A, A orthogonal and B = I plus a nilpotent part

The same question, asked 2 times: 2021 Paper 18*, 25 Jan 2023. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 18

    76
    If A=(1525−2515),B=(10i1),i=−1A =\begin{pmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{- 2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{pmatrix},B =\begin{pmatrix} 1 & 0 \\ i & 1 \end{pmatrix},i=\sqrt{- 1}, and Q=ATBAQ=A^{T}BA, then the inverse of the matrix AQ2021 ATAQ^{2021}{\text{ }A}^{T} is equal to:

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  2. Question 2

    25 Jan 2023

    158
    Let A=[110310−310110]A =\begin{bmatrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{- 3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{bmatrix} and B=[1−i01]B =\begin{bmatrix} 1 & - i \\ 0 & 1 \end{bmatrix}, where i=−1i =\sqrt{- 1}. If M=ATBAM =A^{T}BA, then the inverse of the matrix A2023ATA^{2023}A^{T} is

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