The book is intended for undergraduate and graduate students of Mathematics, Engineering, Social Sciences in general. Professionals and researchers of those fields may also use it as a reference and find the book useful. It is one of the principal branches of Mathematics. The contents are arranged in such a way that a beginner can easily grasp the material step by step. The theories are made lucid through illustrated examples. No prerequisite is necessary for sequential study of the materials, except some preliminary knowledge of calculus for the Calculus part of the book. The 'Generalized Inverse' is essential for undergraduate and graduate study in Statistics, Numerical Analysis, Engineering, Physics and other Social Sciences of most of the institutions of the world. Chapter 1 is on fundamental definitions. It interlinks matrix and linear algebra, tests definiteness of matrices and includes special definitions. Chapters 2 and 3 deal with 'Basic algebraic operations'. 'Square matrices' including some more square matrices normally not available in other texts comprise Chapter 3. Submatrix and partitioned matrix with distinction are in Chapter 4. Chapter 5 includes other matrix operations like Kronecker sums, products, etc., which is normally beyond the contents of many texts. Chapter 6 is on determinants which includes a new method of expansion developed by the author, comparison of methods, the determinant of the product of square and rectangular matrices, factorization of determinants, definiteness tests of matrices, etc., as special features. Chapter 7 is on simultaneous linear algebraic equations with the construction of magic squares. In Chapter 8, the author establishes theproperties of the cofactor matrix, proofs of some properties of the adjoint matrix from the cofactor matrix and introduces classical methods of inversion of nonsingular matrices. Chapter 9 is on elementary operations and matrices similar to other texts with the collection of different symbols from different texts. The material of Chapter 10 is the rank and nullity of a matrix with the development of the Gerstein algorithm from 88's journal and introduces definiteness tests in terms of rank and signature. Chapters 11 and 12 introduce 'Vector Spaces' with the idea of various vector spaces in practice and 'Linear Transformation' with best available collections of materials. Chapters 13 and 14 are on 'Eigenvalues and eigenvectors' and 'Canonical Forms and Matrix Factorization' and include some results available in journals only and not in any texts. Chapter 15 describes linear, bilinear and quadratic forms in details. Chapter 16 to 18 on 'Norms and Measures of Matrices', 'Matrix Calculus' and 'Generalized Inverse' mostly dependent on journal articles and not available in texts of matrices and linear algebra.