A good understanding of matrices and their properties is a necessary prerequisite for progress in almost any field within pure or applied mathematics, for example calculus in several variables, numerical analysis or control theory. This book is based on the course Matrix theory given at Lund University. It starts by recalling the basic theory of matrices and determinants, and then proceeds to more advanced subjects such as the Jordan Normal Form, functions of matrices, norms, normal matrices and singular values. The book may be used for a second course in linear algebra in a bachelor's program in mathematics, or for a first year graduate course in engineering subjects.Distinguishing qualities compared with other texts on the subject are• The book starts gently compared with other texts with the same scope.• There are many carefully worked out examples.• It is possible to use the book for classes on different levels, by selecting parts of the material.The book contains a large number of exercises with answers, indication of difficulty and sometimes hints. The exercises are intended to help the students to think in new ways and to understand the art of proving mathematical statements.