How can I calculate $(4A^{-1} - A^3)^n$?

Question

Let $E= \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and let $A=\begin{bmatrix} I & 0 & 0 \\ 0 & E & I \\ 0 & I & -E \end{bmatrix}$.

How can I calculate $(4A^{-1} - A^3)^n$ for some integer $n$?

Answer 1

If you do block multiplication, calling $$ F=\begin{bmatrix} E & I \\ I & -E \end{bmatrix} $$ you see that $$ A^k=\begin{bmatrix} I & 0 \\ 0 & F^k \end{bmatrix} $$ for any integer $k$, including $-1$.

Thus you need to compute $$ F^3=\begin{bmatrix} (E^2+I)E & E^2+I \\ E^2+I & -(E^2+I)E \end{bmatrix} $$ and, with block reduction, $$ F^{-1}=\begin{bmatrix} (E^2+I)^{-1}E & (E^2+I)^{-1} \\ (E^2+I)^{-1} & -(E^2+I)^{-1}E \end{bmatrix} $$ The key is thus $E^2=I$.