Let A=pmatrix1√10 & 3√10 -3√10 & 1√10pmatrix and B=pmatrix1&-i0&1pmatrix, where i=-1.

Question

Let $A=\begin{pmatrix}\frac1{\sqrt{10}} & \frac3{\sqrt{10}}\ -\frac3{\sqrt{10}} & \frac1{\sqrt{10}}\end{pmatrix}$ and $B=\begin{pmatrix}1&-i\0&1\end{pmatrix}$, where $i=\sqrt{-1}$. If $M=A^TBA$, then the inverse of the matrix $AM^{2023}A^T$ is

Accepted Answer

$A$ is orthogonal: $AA^T=I$. Since $M=A^TBA$, $M^n=A^TB^nA$ for all $n$ (telescoping the $A A^T=I$ pairs).

So $AM^{2023}A^T=A(A^TB^{2023}A)A^T=(AA^T)B^{2023}(AA^T)=B^{2023}$.

$B^n=\begin{pmatrix}1&-ni\0&1\end{pmatrix}$ (by induction), so $B^{2023}=\begin{pmatrix}1&-2023i\0&1\end{pmatrix}$.

Its inverse is $\begin{pmatrix}1&2023i\0&1\end{pmatrix}$.

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