### Abstract:

In this thesis, we consider finite and infinite matrices in linear equations with different structures which arise mainly in the solution of some elliptic partial differential equations in two dimensions. In many of the cases, the solutions lead to infinite systems of linear equations associated with matrices of special structures like diagonal dominance, tridiagonal or having a new sign distribution. The regions considered are either doubly connected or semi infinite. We also consider the theory of finite and infinite tridiagonal matrices, improving some well-known classical results. Nonsingularity criteria are given for matrices with a new sign distribution, which occurs in a conformal mapping problem and viscous fluid flow problem. For the semi infinite region which is bounded on the top by a sloping sinusoidal curve, a theoretical solution in terms of infinite matrices is given leading to numerical evaluation and development of the software. The above problems occur in transmission of electricity in coaxial cables, groundwater flow, conformal mapping, recurrence relations for Bessels functions etc. We also give an error estimate for a finite element method for solution of Laplace's equation resulting in double integrals for physical quantities in applications. The thesis i mainly concerned with using estimates for solving infinite and finite systems with easily computable and meaningful error estimates. The problem in groundwater flow in an infinite region arose from a problem suggested by industry.