The thesis is concerned with the inversion of matrices and the solution of linear systems and eigensystems in a parallel environment. Following an introductory chapter of concepts and definitions in the field of linear algebra, a general survey of parallel machines and algorithms is presented in Chapters 2 and 3, including a detailed description of the Distributed Array Processor (DAP) and the Neptune multiprocessing system. In Chapter 4, a new technique, the double-bordering algorithm, for the solution of linear systems is derived, and its application to the parallel solution of difference systems described. A modified form of the method for the inversion of matrices is derived, implemented on the Neptune multiprocessing system, and its performance compared with that of the Gauss-Jordan and (single) bordering algorithms. The results of the implementation of several other parallel algorithms are also presented. Chapter 5 deals with the class of matrices known as Toeplitz matrices, which arise in the field of signal processing. Trench's algorithm for the inversion of such matrices is implemented on the Neptune multiprocessing system, and, for the solution of banded symmetric Toeplitz systems, the relative efficiencies of three sequential strategies are compared: Levinson's algorithm, the double-bordering algorithm, and a method based on a novel factorisation scheme. Chapter 6 is concerned with the implementation of various iterative methods on the DAP. The solution of several difference systems by the Jacobi, Gauss-Seidel and successive over-relaxation (SOR) algorithms is compared with their solution by a variation (c. 1943) of the algorithms proposed by Hotelling, in which matrix-vector products are replaced by successive matrix squarings. The technique is also applied to the power method for the solution of the eigenvalue problem. The thesis concludes with a summary and recommendations for future work.