Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on the subject by exploring the connections of these tools to matrices. Placing combinatorial and graph-theoretical tools at the forefront of the development of matrix theory, this book uses graphs to explain basic matrix construction, formulas, computations, ideas, and results. It presents material rarely found in other books at this level, including Gersgorin's theorem and its extensions, the Kronecker product of matrices, sign-nonsingular matrices, and the evaluation of the permanent matrix. The authors provide a combinatorial argument for the classical Cayley-Hamilton theorem and a combinatorial proof of the Jordan canonical form of a matrix. They also describe several applications of matrices in electrical engineering, physics, and chemistry.

chapter 1|26 pages


chapter 2|22 pages

Basic Matrix Operations

chapter 3|14 pages

Powers of Matrices

chapter 4|34 pages


chapter 5|12 pages

Matrix Inverses

chapter 6|30 pages

Systems of Linear Equations

chapter 7|32 pages

Spectrum of a Matrix

chapter 8|20 pages

Nonnegative Matrices

chapter 9|26 pages

Additional Topics

chapter 10|24 pages