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Adventures in the world of matrices by Paulus Gerdes

By Paulus Gerdes

The aim of the publication is to offer the reader a sense for the wonder and the surprises of mathematical learn by way of 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 innovations. Then, beginning and experimenting with cycle matrices of low dimensions and fascinating the readers in a variety of actions, attractive effects are bought with beautiful geometrical interpretation. steadily the implications 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 chance to additional use the pc by means of exploring inverses matrices and determinants.

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4. The sum of the numbers in each row and in each column is equal to a+b+c+d+e+f. In the case that the six numbers are different, the matrix is a Latin square. Taking into consideration that the structure of the matrix is similar to the structure of a negative alternating cycle matrix of dimensions 4 by 4 ( alternating cycle matrix of dimensions 6 by 6. ), we can call it a negative ACTIVITY • What are the symmetries of negative alternating cycle matrices of dimensions 6 by 6? Reading the numbers in the first row from left to right we have the sequence (a, b, c, d, e, f), just as when we read the numbers of the last row from right to left.

Reading the numbers in the third row from left to right we have the sequence (c, a, e, b, f, d), the same as when we read the numbers of the fourth row from right to left. In other words, negative alternating cycle 48 Paulus Gerdes matrices of dimensions 6 by 6 present a rotational symmetry of 180o, just like negative alternating cycle matrices of dimensions 4 by 4. 5). 5. 6. 6). When we reflect the matrix across one of these axes the respective positions of the numbers a and f are inverted as are the positions of the numbers b and e and the numbers c and d.

7). The first cycle of the matrix A generated the first cycle of the matrix 3A, where the numbers 6 and 9 alternate; in the second cycle of the matrix 3A the numbers -3 and 15 49 Negative Alternating Cycle Matrices of Dimensions 6 by 6 alternate, and in the third the numbers 12 and 18 alternate. We can say that the matrix 3A is a negative alternating cycle matrix of dimensions 6 by 6. The reader is invited to prove in general that any multiple of a negative alternating cycle matrix of dimensions 6 by 6 constitutes a negative alternating cycle matrix of dimensions 6 by 6 too.

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