By Paulus Gerdes
The aim of the booklet is to provide the reader a sense for the sweetness and the surprises of mathematical examine by means of increase step-by-step a thought of cycle matrices. The notions of matrix and cycle are offered for these readers who don't but comprehend the strategies. Then, beginning and experimenting with cycle matrices of low dimensions and interesting the readers in numerous actions, appealing effects are got with appealing geometrical interpretation. progressively 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 via exploring inverses matrices and determinants.
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Additional info for Adventures in the world of matrices
11. In the previous chapter we saw that there exist alternating cycle matrices of dimensions 4 by 4 that are simultaneously positive and negative. 12. 12. The horizontal and vertical middle lines constitute, together with the diagonals, symmetry axes. In addition the matrices of this form display rotational symmetry of 90o. Rotating the matrix 90o about the center of the matrix places each number where an identical value was. 13 illustrates some examples of corresponding places. 13. 14. 14. 3 5 2 6 -1 4 -1 2 4 3 6 5 5 6 3 4 2 -1 4 -1 6 2 5 3 6 4 5 -1 3 2 3 1 4 0 -5 2 4 3 -5 1 2 0 1 0 3 2 4 -5 -5 4 2 3 0 1 0 2 1 -5 3 4 2 -5 0 4 1 3 Chapter 8 NEGATIVE ALTERNATING CYCLE MATRICES OF DIMENSIONS 6 BY 6 ABSTRACT In chapter 8 negative alternating cycle matrices of dimensions 6 by 6 will be introduced and some of their properties will be analyzed.
Repeat the experience with other pairs of negative alternating cycle matrices of dimensions 6 by 6. Does the same result occur? Chapter 9 MULTIPLICATION OF NEGATIVE ALTERNATING CYCLE MATRICES OF DIMENSIONS 6 BY 6 ABSTRACT In chapter 9 the properties of the multiplication of negative alternating cycle matrices of dimensions 6 by 6 will be analyzed. The concept of positive alternating cycle matrices of dimensions 6 by 6 will be introduced. 1. 1. 2. 2. 2. Do they have alternating cycles? What is the relationship between the two matrices AB and BA?
The question formulated in the last activity will be the theme to be analyzed in the following chapter. Chapter 5 ALTERNATING CYCLE MATRICES OF DIMENSIONS 4 BY 4. 1), resulting from the multiplication of two matrices composed of two alternating cycles, will be analyzed. 1. 1. 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. When we add them, we obtain a matrix with the same structure.
Adventures in the world of matrices by Paulus Gerdes