This is a concise handbook of all the operations that can be performed with and on matrices, with a little bit of set theory and group theory thrown in. It is old-fashioned in that it does not deal at all with linear spaces, and does not give any applications. It is quite thorough on matrix theory and covers much more than a usual introduction would. The present volume is a Dover 2015 corrected reprint of the 1974 fourth revised printing by Longman Group (first edition in 1965).

The chapter on sets and the chapter on groups are flimsy and probably not worth using. The set chapter covers set concepts (mostly from the viewpoint of finite sets), and is mostly about union, intersection, and Venn diagrams. The groups chapter covers the basic definitions, cosets, and has a moderate amount about group representations as subgroups of permutation groups and as matrix groups.

The bulk of the book is on matrices, and it has very thorough coverage, including some uncommon topics such as quadratic forms. The problem is that we no longer study matrices, we study linear algebra, and the book is very weak there. There are no linear spaces, no rank of matrices, no nullspace. There’s nothing about numerical methods. The eigenvalue and diagonalization chapters are good, although the Jordan form is only mentioned and Singular Value Decomposition is not mentioned at all. The handling of systems of linear equations is antiquated even by 1965 standards: Systems are solved either by Cramer’s rule or by matrix inverses, and there’s no mention of Gaussian elimination or LU decomposition.

The book has some value as a matrix reference, and has good exercises (with answers in the back). A much better introductory book is Strang’s *Introduction to Linear Algebra*. It has a linear spaces viewpoint, has a reasonable number of applications, some handling of numeric issues, and has quite a lot on practical methods for solving systems.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.