Inverses and Determinants of n x n Block Matrices

Abstract

Block matrices play an important role in all branches of pure and applied mathematics. In this paper, we study the two fundamental concepts: inverses and determinants of general nxn block matrices. In the first part, the inverses of 2 x 2 block matrices are given, where one of the blocks is a non-singular matrix, a result which can be generalised to a block matrix of any size, by splitting it into four blocks. The second part focuses on the determinants, which is covered in two different methods. In the first approach, we revise a formula for the determinant of a block matrix A, with blocks elements of R; a commutative subring of M-nxn(F). The determinants of tensor products of two matrices are also given in this part. In the second method for computing the determinant, we give the general formula, which would work for any block matrix, regardless of the ring or the field under consideration. The individual formulas for determinants of 2 x 2 and 3 x 3 block matrices are also produced here.