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# Matrix Powers

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Introduction

Matrix Powers

1. Consider the matrix M =  1. Calculate Mn for  = 2, 3, 4, 5, 10, 20, 50.

M2=  M3=  M4=  M5=  M10=  M20=  M50=  1. 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  = 16  P7 =  = 64  S7 =  = 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.

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  , be it negative, rational, whole, will be coherent with the pattern. The values of  however, is limited to only whole numbers. It can go to the extent of whole numbers but it can never be a negative value or a rational value as the error DOMAIN would appear on the calculator. Theoretically, if these disallowed  values were to be solely plugged into the pattern 2n-1  and not Mn , then similar to  the pattern would also function with any rational or negative numbers.
1. 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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