Hardware for executing matrix arithmetic and signal processing algorithms at high speeds is in great demand in many real-time and scientific applications. With the advent of VLSI technology, large numbers of processing elements which cooperate with each other at high speed have become economically feasible. Since any functional error in a high-performance system may seriously jeopardize the operation of the system and its data integrity, some level of fault tolerance must be incorporated in order to ensure that the results of long computations are valid. Since the major computational requirements for many important real-time signal processing tasks can be reduced to a common set of basic matrix operations, the development of a unified fault-tolerant scheme for matrix operations can solve the problems of both reliable signal processing and reliable matrix operations. Earlier work proposed a low-cost checksum scheme for fault-tolerant matrix operations on multiple processor systems. However, this scheme can only correct errors in matrix multiplication; it can detect, but not correct, errors in matrix-vector multiplication, LU decomposition, matrix inversion, etc. In orderto solve these problems with the checksum scheme, a very general matrix encoding scheme is proposed in this paper to achieve fault-tolerant matrix arithmetic and signal processing with linear arrays, which are believed to hold the most promise in VLSI computing structures for their flexibility, low cost, and applicability to most of the interesting algorithms. This proposed technique is, therefore, a very cost-effective encoding technique to achieve fault-tolerant matrix arithmetic and signal processing on highly concurrent VLSI computing structures.