By A A Pankov

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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? Reading the numbers in the first row from left to right we have the sequence (a, b, c, d, e, f), just as when we read the numbers of the last row from right to left.

10). 10. We have proved that the following is valid under multiplication for matrices consisting of the given structures: x = . ACTIVITY • Show that for alternating cycle matrices with the given structures the following holds under multiplication: x = . When we summarize the main conclusions of chapters 4 and 5, the following is valid for the structures of alternating cycle matrices when they are multiplied: x = x = and . 11 joins the four results in the multiplication table for alternating cycle matrices of dimensions 4 by 4.

12). 12. 1. 13) where p = ae+bf+cg+dh, q = ah+bg+cf+de, r = af+bh+ce+dg and s = ag+be+ch+df. 13. 14). 14. Let us now observe another particularity. When we introduced the number p by p = ae+bf+cg+dh, we had for the elements on the principal diagonal: p = ae+bf+cg+dh = bf+dh+ae+cg = cg+ae+dh+bf = dh+cg+bf+ae. In the first element, that is ae+bf+cg+dh, the part ae appears in the first place (1); in the second element, that is bf+dh+ae+cg, the part ae appears in the third place (3); in the third element, the part ae appears in the second place (2) and finally, in the fourth element, the part ae appears in the fourth place (4).