By Paulus Gerdes
The aim of the e-book is to offer the reader a sense for the wonder and the surprises of mathematical study through increase step-by-step a concept of cycle matrices. The notions of matrix and cycle are offered for these readers who don't but understand the thoughts. Then, beginning and experimenting with cycle matrices of low dimensions and interesting the readers in a number of actions, attractive effects are got with beautiful geometrical interpretation. progressively 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 chance to additional use the pc by way of exploring inverses matrices and determinants.
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P s r q r p q s s q p r q r s p x t w v u v t u w w u t v u v w t = ? 5. ACTIVITIES • Calculate, one by one, the elements of the resulting matrix. Compare its elements and analyze the structure of the matrix resulting from the multiplication. 6. 6. The elements in the principal diagonal are equal: pt+rw+sv+qu = sv+pt+qu+rw = rw+qu+pt+sv = qu+sv+rw+pt. The elements in the secondary diagonal are equal: pu+rv+sw+qt = sw+pu+qt+rv = rv+qt+pu+sw = qt+sw+rv+pu. The other elements of the matrix constitute an alternating cycle, as we have: pv+rt+su+qw = su+pv+qw+rt = rt+qw+pv+su = qw+su+rt+pv and pw+ru+st+qv = st+pw+qv+ru = ru+qv+pw+st = qv+st+ru+pw.
Is this a generalization or will it only result from particular numbers that appear as elements of those two matrices? ACTIVITY • Analyze, in general, the multiplication of alternating cycle matrices where one presents the structure and the other the structure . Let us analyze in the first place the situation where the first matrix has the structure and the second matrix the structure any numbers. 6), where p, q, r, s, a, b, c and d denote 34 Paulus Gerdes p s r q r p q s s q p r q r s p x a b c d b d a c c a d b d c b a = ?
3 illustrates the matrix and the three cycles at the same time. 3. 2 a). In the first row a 2 and a 3 appear once. In the second row from left to right, a 3 and then a 2 appears once. In the third row a 2 and then a 3 each appear once. The pattern continues. A similar pattern can be noted with the columns. In the first column a 2 and then a 3 appears once. In the second column from top to bottom a 3 and then a 2 each appear once. In the third column first a 2 and then a 3 each appear once. This pattern also continues.