By Paulus Gerdes

The aim of the e-book is to offer the reader a sense for the sweetness and the surprises of mathematical learn via increase step-by-step a thought of cycle matrices. The notions of matrix and cycle are awarded for these readers who don't but understand the options. Then, beginning and experimenting with cycle matrices of low dimensions and interesting the readers in numerous actions, appealing effects are bought with appealing geometrical interpretation. progressively the consequences are prolonged to cycle matrices of upper dimensions after which generalised. A be aware is integrated for these readers who want to use a working laptop or computer in exploring cycle matrices. appendices supply the prospect to additional use the pc by means of exploring inverses matrices and determinants.

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**Example text**

8). t w v u v t u w w u t v u v w t x p s r q r p q s s q p r q r s p = ? 8. In agreement with the theorem that we just proved, we know already that the structure of the matrix resulting from the multiplication is the same as that of the matrices to be multiplied. Therefore we need to calculate only the four elements of the first row in order to know all elements of the matrix resulting from the multiplication. The first element of the first row is equal to tp+vs+wr+uq. 6): pt+rw+sv+qu. The two elements are equal as the order of the addition of the parts is free and the multiplication of the numbers is commutative (pt = tp, vs = sv, wr = rw, uq = qu): tp+vs+wr+uq = tp+wr+vs+uq = pt+rw+sv+qu.

8 illustrates some examples. 8. After this analysis of the symmetries of negative alternating cycle matrices of dimensions 4 by 4, the reader is invited to produce a similar analysis of positive alternating cycle matrices of the same dimensions. ACTIVITIES • Find the symmetries of positive alternating cycle matrices of dimensions 4 by 4 • ( ). Do alternating cycle matrices exist that are simultaneously negative and positive? If yes, what symmetries do those matrices have? 9). 9. In general, this matrix does not have symmetry axes.

Add pairs of the constructed matrices and analyze the resulting matrices. Multiply pairs of matrices of the considered type and study the structure of the matrices resulting from the multiplication. 1. When we add them, we obtain a matrix with the same structure. 2. -1 3 2 5 5 2 3 -1 2 -1 5 3 + -2 -3 1 4 1 -2 4 -3 -3 4 -2 1 4 1 -3 -2 = 1 2 0 6 0 1 6 2 2 6 1 0 6 0 2 1 26 Paulus Gerdes Let us prove that this is true in general. 3). Immediately one sees that the matrix resulting from the addition has the same cyclic structure.