The purpose of the book is to give the reader a feeling for the beauty and the suprises of mathematical research by building up step by step a theory of cycle matrices. The notions of matrix and cycle are presented for those readers who do not yet know the concepts. Then, starting and experminenting with cycle matrices of low dimensions and engaging the readers in various activities, beautiful results are obtained with attractive geometrical interpretation. Gradually the results are extended to cycle matrices of higher dimensions and then generalized. A note is included for those readers who would like to use a computer in exporing cycle matrices. Two appendices give the possibility to further use the computer by exploring inverses matrices and determinants.