This book gives an elementary introduction to matrix computation and it also includes some new results obtained in recent years. This book consists of nine chapters. It includes Gaussian elimination, classical iterative methods and Krylov subspace methods for solving linear systems; the rounding error analysis of Gaussian elimination; the perturbation analysis of linear systems; orthogonal decomositions for solving linear least squares problems; and some classical methods for eigen-problems. In the last chapter, a brief survey of the latest developments in using boundary value methods for solving initial value problems of ordinary differential equations is given. This is a textbook for the senior students majoring in scientific computing and information science. It will also be useful to all who teach or study the subject.