Properties of Block Matrices

In this master thesis, we study the block matrices and their properties. After giving a general overview on matrices, block matrices, different types of block matrices, and multiplication of two block matrices are discussed. In the inverse section, we first examine inverses of 2×2 block diagonal and block triangular matrices, ideas of proofs here can be extended to a general n × n block diagonal or a block triangular matrix. Then we give the inverse formula for 2 × 2 block matrix, in the case that one of the blocks is invertible. We then generalise this to any n×n block matrix by splitting it into 4 blocks (by producing a 2×2 block matrix). Determinant chapter is covered by two different methods, existing in the literature. First we revise a formulae for determinant of a block matrix where the blocks (matrices) belong to a commutative subring of Mn×n(F), where F is a field or a commutative ring. Then we give the general formula which would work for any block matrix, without any commutativity condition between the blocks. We also present formulas for the determinant of tensor product of two given matrices. Keywords: block matrix, inverses, determinants, tensor products