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Matrices Portfolio

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Matrix powers

A matrix is a rectangular array of numbers (or letters) arranged in rows and columns. These numbers (or letters) are known as entries. Entries can be added and multiplied, but also squared. The aim of this portfolio is to investigate squaring matrices.

When we square the matrix M = image00.pngimage00.png what we receive is a) image40.pngimage40.png = image51.pngimage51.png = image61.pngimage61.png.

Calculating the matrices for n = 3, 4, 5, 10, 20 and 50:

image71.pngimage71.png= image01.pngimage01.png

image13.pngimage13.png= image24.pngimage24.png

image35.pngimage35.png= image39.pngimage39.png

image41.pngimage41.png= image42.pngimage42.png

image43.pngimage43.png= image44.pngimage44.png

image45.pngimage45.png= image46.pngimage46.png

Examples shown above clearly indicate that while zero entries remain the same, non-zero entries change. Each entry is raised to a given power separately. Raising them to any power does not change the zero-entries.

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These matrices are simplified to 2image56.pngimage56.png and 2image60.pngimage60.png to make it easier to notice any patterns. Calculating Pn and Sn 

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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.

image22.pngimage22.png = image23.pngimage23.png = image25.pngimage25.png = 65536image26.pngimage26.png

image27.pngimage27.png = image28.pngimage28.png = image29.pngimage29.png

image30.pngimage30.png = image31.pngimage31.png = image32.pngimage32.png = 4096image33.pngimage33.png

image34.pngimage34.png = image36.pngimage36.png = image37.pngimage37.png

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.

image38.pngimage38.png 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.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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