By Paulus Gerdes

The aim of the publication is to provide the reader a sense for the sweetness 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 provided for these readers who don't but comprehend the ideas. Then, beginning and experimenting with cycle matrices of low dimensions and interesting the readers in a variety of actions, appealing effects are bought with appealing geometrical interpretation. progressively the implications are prolonged to cycle matrices of upper dimensions after which generalised. A word is integrated for these readers who wish 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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Example text

2. 2. Do they have alternating cycles? What is the relationship between the two matrices AB and BA? What can be said about the matrix AB+BA ? The principal diagonals of the two matrices are equal and on both diagonals the number 47 is repeated. In the same way, the secondary diagonals are equal and their elements are all equal to 8. 3). In the second matrix, there are also two cycles in which the same numbers appear as in the first matrix, but the cycles are different in phase. In other words, the numbers in the alternating cycles of the second matrix are inverted relative to the numbers in the alternating cycles of the first matrix.

Let us observe now another particularity. When we introduce the number a through a = pt+rw+sv+qu, we had for the elements in the principal diagonal a = pt+rw+sv+qu = sv+pt+qu+rw = rw+qu+pt+sv = qu+sv+rw+pt. In the first element, pt+rw+sv+qu, the part pt appears in the first place (1); in the second element, that is sv+pt+qu+rw, the part pt appears in the second place (2); in the third element, the part pt appears in the third place (3) and finally, in the fourth element, the part pt appears in the fourth place (4).

Let p = ae+bf+cg+dh. 9 four times the number p. The elements on the secondary diagonal are equal as one may change the order of the parts to be added: ah+bg+cf+de = bg+de+ah+cf = cf+ah+de+bg = de+cf+bg+ah Let q = ah+bg+cf+de. 9 four times the number q. 11). 11. Let r be the element in the first row and the second column, that is, r = af+bh+ce+dg. As one may change the order of the four parts without changing the value of their sum, we have: r = af+bh+ce+dg = bh+dg+af+ce = ce+af+dg+bh = dg+ce+bh+af.