Perhaps you can   consider   A Combinatorial Approach to Matrix Theory and Its Applications (Discrete Mathematics and Its Applications) Hardcover – August 6, 2008

by Richard A. Brualdi (Author), Dragos Cvetkovic (Author).  This relatively elementary book.

In the book A Combinatorial Approach to Matrix Theory and Its Applications the authors employ 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.

''After reviewing the basics of graph theory, elementary counting formulas, fields, and vector spaces, the book explains the algebra of matrices and uses the König digraph to carry out simple matrix operations. It then discusses matrix powers, provides a graph-theoretical definition of the determinant using the Coates digraph of a matrix, and presents a graph-theoretical interpretation of matrix inverses. The authors develop the elementary theory of solutions of systems of linear equations and show how to use the Coates digraph to solve a linear system. They also explore the eigenvalues, eigenvectors, and characteristic polynomial of a matrix; examine the important properties of nonnegative matrices that are part of the Perron–Frobenius theory; and study eigenvalue inclusion regions and sign-nonsingular matrices. The final chapter presents applications to electrical engineering, physics, and chemistry.''

Thanks Prof.Miodrag. Your answer was very useful to me. I was searching a book like that..