Products and inverses of multidiagonal matrices with equally spaced diagonals

Let $n,k$ be fixed natural numbers with $1\le k\le n$ and let $A_{n+1,k,2k,\dots,sk}$ denote an $(n+1)\times (n+1)$ complex multidiagonal matrix having $s=[n/k]$ sub- and superdiagonals at distances $k,2k,\dots,sk$ from the main diagonal. We prove that the set $\mathcal{MD}_{n,k}$ of all such multidiagonal matrices is closed under multiplication and powers with positive exponents. Moreover the subset of $\mathcal{MD}_{n,k}$ consisting of all nonsingular matrices is closed under taking inverses and powers with negative exponents. In particular we obtain that the inverse of a nonsingular matrix $A_{n+1,k}$ (called $k$-tridigonal) is in $\mathcal{MD}_{n,k}$, moreover if $n+1\le 2k$ then $A^{-1}_{n+1,k}$ is also $k$-tridigonal. Using this fact we give an explicite formula for this inverse.