T h e matrix calculus is widely applied nowadays in various branches of mathematics, mechanics, theoretical physics, theoretical electrical engineer­ ing, etc. However, neither in the Soviet nor the foreign literature is there a book that gives a sufficiently complete account of the problems of matrix theory and of its diverse applications. The present book is an attempt to fill this gap in the mathematical literature.The book is based on lecture courses on the theory of matrices and its applications that the author has given several times in the course of the last seventeen years at the Universities of Moscow and Tiflis and at the Moscow Institute of Physical Technology.The book is meant not only for mathematicians (undergraduates and research students) but also for specialists in allied fields (physics, engi­ neering) who are interested in mathematics and its applications. Therefore the author has endeavoured to make his account of the material as accessible as possible, assuming only that the reader is acquainted with the theory of determinants and with the usual course of higher mathematic? within the programme of higher technical education.