Hence, we can observe that the general expression for the matrix = =
When considering the matrices P = and S = , more conclusions can be drawn.
Raising each of them to the second power gives:
= = = 2
= = = 2
These matrices are simplified to 2 and 2 to make it easier to notice any patterns. Calculating Pn and Sn for other values of n we obtain:
= = = 2 = 4
= = = 2 = 4
= = = 2 = 16
= = = 2 = 16
= = = 512
= = = 512
Analysis of above pattern shows that:
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dividing the first result by 2 raised to the power 1 less than the number to which the matrix is raised (e.g. P10; number 2 will be raised to 10 – 1 = 9 power, which gives 512), results in a new matrix with two pairs of similar entries situated diagonally;
- the difference between two entries in one row or one column is equal 2, therefore we can either add or subtract 1 from their mean to get all entries of one matrix.
Matrix is the general expression for the pattern described above. Matrices M, P and S in this portfolio can be used as examples, where n = 1, 2 and 3.
Summarising:
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k is the arithmetic mean of two entries found in one row or column
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k can be both odd or even
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k does not have to be an integer number
= = 8
-
k does not have to be a positive number
= = 16
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n is the power to which the matrix is raised
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to obtain the entries where the difference equals 2 (in a row or a column) we have to divide each of them by 2n – 1
= =
When technology was used to investigate what happened with further values of k and n it occurred, that for greater numbers the pattern was not true. In the case of, e.g. k = 1500 and n = 2, the pattern worked. Increasing n to 3, however, caused all the entries to be the same.
This was also checked for the matrices P and S.
= = = 65536
= =
= = = 4096
= =
The pattern works as long as the results are less than 10 billions. If they exceed this number, all the entries will be exactly the same.
10 000 000 000
Therefore, this does not seem to be true for every number.
The results hold true in general because in real life situations so large numbers are not frequently used. For smaller numbers the pattern fits thoroughly.
