An introduction to matrices and quadratic forms for second-year degree modules in linear algebra. Building on both skills and knowledge attained during A Level and earlier degree courses, it discusses topics such as the reduction of suitable matrices to diagonal matrices by means of non-singular or orthogonal matrices constructed with eigenvectors. Examples and exercises are used as teaching aids throughout and ideas for investigation and project work help to place the subject in context. The text seeks to enhance student motivation and learning by the inclusion of historical contexts, real life situations and the discussion of links with other areas of mathematics.