# Matrix Powers

Extracts from this document...

Introduction

Muhammad Faiz M.Ismail 12B

Matrix Powers

- Consider the matrix M =

- Calculate Mn for = 2, 3, 4, 5, 10, 20, 50.

M2=

M3=

M4=

M5=

M10=

M20=

M50=

- Describe in words any pattern you observe.

Using a TI-83 Plus to calculate each of these power matrices, we are able to find the th matrix. The matrix provided is an identity matrix, which is uniquely defined by the property In M = M thus the value of Mn is equated to two to the power of as well; the value zero stays constant for other values of . The matrix Mn can also be calculated using geometric sequence: Un = U1rn-1;as the common ratio between each matrix is a factor of two.

Middle

P7 = = 64S7 = = 64

P10 = = 512

S10 = = 512

From a standard matrix , we can see that the difference between and , and is two. Therefore we could infer that it follows the pattern , and that in matrix P is two whereas in matrix S is three. However, a coefficient had to be factorised in order to have that difference of two. A pattern for the coefficient was found as 2n-1.

- Now consider matrices of the form .

Steps 1 and 2 contain examples of these matrices for = 1, 2 and 3.

Consider other values of , and describe any pattern(s)

Conclusion

- Explain why your results hold true in general.

Since we have proven the general expression for Mn and all those examples are part of the pattern, the results holds true for all examples in the pattern except for the restrictions that have been proven .

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

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