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By Paulus Gerdes

The aim of the e-book is to provide the reader a sense for the sweetness and the surprises of mathematical study by means of increase step-by-step a conception of cycle matrices. The notions of matrix and cycle are awarded for these readers who don't but recognize the strategies. Then, beginning and experimenting with cycle matrices of low dimensions and interesting the readers in a number of actions, attractive effects are received with beautiful geometrical interpretation. progressively the implications 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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As the structure is similar to that of positive alternating cycle matrices of dimensions 4 by 4 ( ), we can say that the products AB and BA constitute positive alternating cycle matrices of dimensions 6 by 6. Besides the constant diagonals, two alternating cycles appear instead of only one, as was the case of positive alternating cycle matrices of dimensions 4 by 4. 4 presents the general form and the cyclic structure of positive alternating cycle matrices of dimensions 6 by 6.

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 presents the cycle composed by the numbers 3 and 4. 2). 2. 3). 3. 4 has the same cyclic structure. 4. 5. 5. 4 2 3 1 1 3 2 4 2 1 4 3 + -1 5 -2 3 5 3 -1 -2 -2 -1 3 5 3 -2 5 -1 = 2 9 -1 5 9 5 2 -1 -1 2 5 9 5 -1 9 2 19 Alternating Cycle Matrices of Dimensions 4 by 4. 6). 6. ACTIVITIES • • How can one characterize the matrices resulting from the multiplication?

3b). Will all matrices thus constructed be Latin squares? Let a, b, c, d, e, and f be any six numbers. 4 illustrates. 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?

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