Products and inverses of multidiagonal matrices with equally spaced diagonals

Let n; k be fixed natural numbers with 1 ? n and let An+1,k,2k,...,sk denote an (n+1)x(n+1) complex multidiagonal matrix having s = [n/k] sub- superdiagonals at distances k, 2k,...,sk from the main diagonal. We prove that set MDn,k of all such matrices is closed under multiplication powers positive exponents. Moreover subset consisting nonsingular taking inverses negative In particular we obtain inverse a An+1,k (called k-tridigonal) inMDn;k, moreover if n+1 2k then A-1 n+1,k also k-tridigonal. Using this fact give explicit formula for inverse.