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Today, in almost all disciplines of SET (Science, Engineering and Technology), an extensive course on the Theory of Matrices has become indispensable part of the curriculum. Book is designed for a first course on Matrix Theory for graduate level. This book fulfills the freshman's need for a basic study for the concepts, methods and logic. It should also serve non-mathematical majors as an inviting introduction to modern mathematical methods. The systematic treatment of matrices and the detailed solution form a concrete foundation for advanced study of mathematical concepts. Other special features are theorems, proved in full. The coverage of the book is quite extensive, a large number of chapter-end problems are incorporated, answers to almost all the problems are provided to help the student in self study. About 200 worked examples expertly woven into the text and useful sets of problems interpolated at the end of all chapters/sections. The book addresses the conventional topics: matrices, inverse of matrices, adjoint of matrices, rank of matrices, Eigen values and Eigen vectors, inner product of vectors, Quadratic forms, companion matrix and location of zeros- with symmetry.