Matrix algebra plays a very important role in statistics and in many other dis- plines. In many areas of statistics, it has become routine to use matrix algebra in thepresentationandthederivationorveri?cationofresults. Onesuchareaislinear statistical models; another is multivariate analysis. In these areas, a knowledge of matrix algebra isneeded in applying important concepts, as well as instudying the underlying theory, and is even needed to use various software packages (if they are to be used with con?dence and competence). On many occasions, I have taught graduate-level courses in linear statistical models. Typically, the prerequisites for such courses include an introductory (- dergraduate) course in matrix (or linear) algebra. Also typically, the preparation provided by this prerequisite course is not fully adequate. There are several r- sons for this. The level of abstraction or generality in the matrix (or linear) algebra course may have been so high that it did not lead to a “working knowledge” of the subject, or, at the other extreme, the course may have emphasized computations at the expense of fundamental concepts. Further, the content of introductory courses on matrix (or linear) algebra varies widely from institution to institution and from instructor to instructor. Topics such as quadratic forms, partitioned matrices, and generalized inverses that play an important role in the study of linear statistical models may be covered inadequately if at all. From a review: THE AUSTRALIAN AND NEW ZEALAND JOURNAL OF STATISTICS "This is a book that will be welcomed by many statisticians at most stages of professional development. ...It is essentially a carefully sequenced and tightly interlocking collections of proofs in an elementary, though very pure mathematical style." This book presents matrix algebra in a way that is well-suited for those with an interest in statistics or a related discipline. It provides thorough and unified coverage of the fundamental concepts along with the specialized topics encountered in areas of statistics such as linear statistical models and multivariate analysis. It includes a number of very useful results that have only been available from relatively obscure sources. Detailed proofs are provided for all results. The style and level of presentation are designed to make the contents accessible to a broad audience. The book is essentially self-contained, though it is best-suited for a reader who has had some previous exposure to matrices (of the kind that might be acquired in a beginning course on linear or matrix algebra). It includes exercises, it can serve as the primary text for a course on matrices or as a supplementary text in courses on such topics as linear statistical models or multivariate analysis, and it willbe a valuable reference. David A. Harville is a research staff member emeritus in the Mathematical Sciences Department of the IBM T.J. Watson Research Center. Prior to joining the Research Center, he spent ten years as a mathematical statistician in the Applied Mathematics Research Laboratory of the Aerospace Research Laboratories (at Wright-Patterson, Air Force Base, Ohio), followed by twenty years as a full professor in the Department of Statistics at Iowa State University. He has extensive experience in the area of linear statistical models, having taught (on numberous occasions) M.S.- and Ph.D.-level courses on that topic, having been the thesis adviser of ten Ph.D. students, and having authored more than 70 research articles. His work has been recognized by his having been named a Fellow of the American Statistical Association and of the Institute of Mathematical Statistics, by his election as a member of the International Statistical Institute, and by his having servedas an associate editor of Biometrics and of the Journal of the American Statistical Association . David A. Harville is a research staff member in the Mathematical Sciences Department of the IBM T.J.Watson Research Center. Prior to joining the Research Center he spent ten years as a mathematical statistician in the Applied Mathematics Research Laboratory of the Aerospace Research Laboratories (at Wright-Patterson, FB, Ohio, followed by twenty years as a full professor in the Department of Statistics at Iowa State University. He has extensive experience in the area of linear statistical models, having taught (on numberous occasions) M.S.and Ph.D.level courses on that topic,having been the thesis adviser of 10 Ph.D. students,and having authored over 60 research articles. His work has been recognized by his election as a Fellow of the American Statistical Association and the Institute of Mathematical Statistics and as a member of the International Statistical Institute and by his having served as an associate editor of Biometrics and of the Journal of the American Statistical Association.