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# Matrices Portfolio

Extracts from this document...

Introduction

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 =  what we receive is a)  =  = .

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

=

=

=

=

=

=

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.

Middle

These matrices are simplified to 2 and 2 to make it easier to notice any patterns. Calculating Pn and Sn

Conclusion

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.

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

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