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Adventures in the world of matrices by Paulus Gerdes

By Paulus Gerdes

The aim of the publication is to provide the reader a sense for the sweetness and the surprises of mathematical study via increase step-by-step a idea of cycle matrices. The notions of matrix and cycle are awarded for these readers who don't but comprehend the strategies. Then, beginning and experimenting with cycle matrices of low dimensions and fascinating the readers in quite a few actions, appealing effects are got with appealing geometrical interpretation. steadily the consequences are prolonged to cycle matrices of upper dimensions after which generalised. A observe is incorporated 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 way of exploring inverses matrices and determinants.

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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.

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. 3. 3). 3. Surprisingly, the matrices resulting from the two multiplications are equal. They also present the same cyclic structure as the two initial matrices. Is this an exceptional case? 4). 4. Once more, the matrices resulting from the two multiplications are equal and display the same structure as the two initial matrices.

Third Part: 37 ACTIVITIES • • Will it be possible to picture alternating cycle matrices of dimensions 6 by 6? When possible, analyze if it is possible to distinguish them into two types, negative and positive. Try to construct negative and positive alternating cycle matrices of dimensions 8 by 8. Experiment with multiplying pairs of those matrices and analyze the structure of the products obtained in this way. Does there exist some parallelism with the situation we have already studied of negative and positive alternating cycle matrices of dimensions 4 by 4?

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