By Paulus Gerdes
The aim of the booklet is to offer the reader a sense for the wonder and the surprises of mathematical examine by way of increase step-by-step a conception of cycle matrices. The notions of matrix and cycle are offered for these readers who don't but understand the suggestions. Then, beginning and experimenting with cycle matrices of low dimensions and interesting the readers in a number of actions, appealing effects are bought with appealing geometrical interpretation. progressively the consequences are prolonged to cycle matrices of upper dimensions after which generalised. A word is integrated for these readers who want to use a working laptop or computer in exploring cycle matrices. appendices provide the prospect to extra use the pc via exploring inverses matrices and determinants.
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Additional info for Adventures in the world of matrices
That is, what particularity does the square of the matrix display? 16 shows an example of the multiplication of a cycle matrix by itself. One notes that in the cycle the number -7 is repeated. 16. Let us analyze this question in general. 18. 17. 18. 19. 19. 12 and multiply pairs of these matrices. Will it be possible to note some interesting particularity of the matrices resulting from the multiplication? If yes, experiment with some pairs and try to prove a general result. The question formulated in the last activity will be the theme to be analyzed in the following chapter.
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. It appears that the moment to analyze the general case has come. Alternating Cycle Matrices of Dimensions 4 by 4. 5), where p, q, r, s, t, u, v, and w are any numbers. 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 = ?
2e). Negative Alternating Cycle Matrices of Dimensions 6 by 6 47 By consequence, we can conclude that, like we had seen in the case of cycle matrices of dimensions 4 by 4, each of the six numbers appears exactly once in each row and once in each column. The considered matrix is a Latin square and the sum of the numbers in each row and in each column is constant (= 19). 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.
Adventures in the world of matrices by Paulus Gerdes