Given $$E = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ define the following block matrix $$A = \begin{bmatrix} I & 0 & 0 \\ 0 & E & I \\ 0 & I & -E \end{bmatrix}$$ Determine $\left( 4 A^{-1} - A^3 \right)^n$ for all $n \in \mathbb{N}$.
I am stuck even at determining the inverse $A^{-1}$. What I normally do is use the Gauss-Jordan elimination or, for $2 \times 2$ matrices, I use the known identity. But I have absolutely no idea what to do with block matrices. I could write the entire block matrix explicitly and work with that, but that would be very painful to do. I am sure there is some elegant way to work with block matrices, which I hope someone would explain to me.
$E^2 = I$
$A^{-1} = \begin{bmatrix} I\\&\frac 12 E & \frac 12 I\\&\frac 12 I &-\frac 12 E\end{bmatrix}$
However, it is not usually this easy.
As $(4A^{-1} - A^3) = A^{-1}(4I - A^4),$ I would be inclined to inspect $(4I - A^4)$
$4I - A^4$ gives a diagonal matrix with $3$ in the first $3$ entries and $0$ everywhere else.