This book is written as a textbook for undergraduate students familiar with linear algebra and abstract algebraic structures. It could be used as an excellent textbook for a one semester course at university and it will prepare students for a graduate course on Lie groups, Lie algebras, etc. The author begins with basic facts on matrices (definition, operations and matrices as linear transformations), quaternions, general linear groups and change of basis. In the eight following chapters he explains matrix groups, orthogonal groups, topology of matrix groups, Lie algebras, matrix exponentiation, matrix groups as manifolds, Lie brackets and maximal tori. The book combines an intuitive style of writing (with many examples and a geometric motivation) with rigorous definitions and proofs, giving examples from fields of mathematics, physics and other sciences, where matrices are successfully applied. The book will surely be interesting and helpful for students of algebra and their teachers.