This paper deals with the execution of matrix operations and with the solution of linear systems in Reconfigurable Processor Arrays (RPAs). RPAs belong to the category of SIMD massively parallel computers; their distinctive feature is the reconfigurable interconnection network, which allows to dynamically adapt the network topology to the communication requirements of the running algorithms. The paper shows that RPAs support the optimum execution of matrix operations, such as sum, product and inverse, as well as the optimum solution of dense linear systems by means of the Gaussian elimination method. Algorithms are presented for the selected set of elementary matrix operations as well as for Gaussian elimination, considering the case in which the size of the matrices involved in the computation equals the size of the reconfigurable processor arrays. Partitioned algorithms are also proposed to deal with the more general case in which the size of the matrices involved is larger than the size of the reconfigurable processor array. All the algorithms presented are shown to be optimum, in the sense that they achieve a speed up of O(N), over the sequential implementation, using O(N) processors.