This book is designed to be used either as a textbook for a formal course in matrix theory or as a very supplement to all current linear algebra texts. Every chapter begins with a clear statement of pertinent definitions, theorems together with illustrative and other descriptive material. This is followed by graded sets of solved problems with some unsolved exercises at the end of each section. Topics covered include vectors, spectrum of eigenvalues of different matrices, quadratic forms followed by matrix inequalities. Considerably more solved problems has been included in this edition. This has been done to make the book more flexible, to prove a more useful book of reference for clear understanding of topics discussed here and to stimulate further interest in the topics. This book is primarily intended for use as a textbook at the graduate level and in particular for students pursuing masters in mathematics. The material is in the most simpleminded straight forward manner, organized so that the book can also be used in a wider setting within today’s modern university and competitive exams. It is better that students learn to organize and write proofs themselves than to cover more subject matter.