By Paulus Gerdes

The aim of the ebook is to provide the reader a sense for the wonder and the surprises of mathematical learn via build up step-by-step a idea of cycle matrices. The notions of matrix and cycle are offered for these readers who don't but understand the strategies. Then, beginning and experimenting with cycle matrices of low dimensions and fascinating the readers in a variety of actions, attractive effects are acquired with appealing geometrical interpretation. steadily the consequences are prolonged to cycle matrices of upper dimensions after which generalised. A be aware is incorporated for these readers who want to use a working laptop or computer in exploring cycle matrices. appendices provide the prospect to additional use the pc via exploring inverses matrices and determinants.

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3. ACTIVITY • Construct some Latin squares of dimensions 4 by 4 that are not alternating cycle matrices. Let us now analyze the alternating cycle matrices, of dimensions 4 by 4, from the point of view of the symmetries they present. Can these matrices display rotational symmetry? Can they have axes of symmetry? ACTIVITIES • Analyze which symmetries negative alternating cycle matrices of dimensions 4 by 4 have ( )? More Properties of Alternating Cycle Matrices of Dimensions 4 by 4 41 • Find the symmetries that positive alternating cycle matrices of dimensions 4 by 4 • display ( )?

9. 3]). Alternating Cycle Matrices of Dimensions 4 by 4. 9. 10. ACTIVITY • Analyze in the same way the order of the parts in the elements equal to b, c, and d, respectively. Will the same surprising particularity be observed? 10 ( ) are very much related. 1 ( one type we arrive easily at a matrix of the other type and vice-versa. ACTIVITIES • • Add matrices of the two distinct types. Does something special happen? Multiply matrices of the two distinct types. Does something special happen? Will it be possible to discover some multiplication rule?

12 shows us the table (of signs) of multiplication of numbers: ‘negative’ times ‘negative’ gives ‘positive’, etc. 12. The two tables are very similar. This led me to introduce names for the two types of alternating cycle matrices that allows us to remember the respective multiplication table. We may call the alternating cycle matrices with the structure cycle matrices with the structure In other words, negative and the alternating positive. 1. negative alternating cycle matrices of dimensions 4 by 4 have the structure two alternating cycles; 2.