The inverse of banded matrices

The inverses of r-banded matrices, for r=1,2,3 have been thoroughly investigated as one can see from the references we provide. Let B"r","n ([email protected][email protected]?n) be an nxn matrix of entries {a"j^i}, [email protected][email protected]?r, [email protected][email protected]?r, with the remaining un-indexed entries all zeros. In this paper, generalizing a method of Mallik (1999) [5], we give the LU factorization and the inverse of the matrix B"r","n (if it exists). Our results are valid for an arbitrary square matrix (taking r=n), and so, we will give a new approach for computing the inverse of an invertible square matrix. Our method is based on Hessenberg submatrices associated to B"r","n.