If A=pmatrix1√5&2√5-2√5&1√5pmatrix, B=pmatrix1&0&1pmatrix, i=-1, and Q=A^TBA, then the inv…

Question

If $A=\begin{pmatrix}\frac1{\sqrt5}&\frac2{\sqrt5}\-\frac2{\sqrt5}&\frac1{\sqrt5}\end{pmatrix}$, $B=\begin{pmatrix}1&0\i&1\end{pmatrix}$, $i=\sqrt{-1}$, and $Q=A^TBA$, then the inverse of the matrix $AQ^{2021}A^T$ is equal to

Accepted Answer

$AA^T=I$ (orthogonal), so $Q^n=A^TB^nA$ (telescoping), and $AQ^{2021}A^T=(AA^T)B^{2021}(AA^T)=B^{2021}$.

$B^n=\begin{pmatrix}1&0\ni&1\end{pmatrix}$, so $B^{2021}=\begin{pmatrix}1&0\2021i&1\end{pmatrix}$, whose inverse is $\begin{pmatrix}1&0\-2021i&1\end{pmatrix}$.

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