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Matrix Powers

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Introduction

Muhammad Faiz M.Ismail 12B

Matrix Powers

  1. Consider the matrix M = image00.pngimage00.png
  1. Calculate Mn for image04.pngimage04.png = 2, 3, 4, 5, 10, 20, 50.

M2= image35.pngimage35.png

M3= image45.pngimage45.png

M4= image56.pngimage56.png

M5= image01.pngimage01.png

M10= image08.pngimage08.png

M20= image14.pngimage14.png

M50= image24.pngimage24.png

  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 image04.pngimage04.pngth 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 image04.pngimage04.pngas well; the value zero stays constant for other values of image04.pngimage04.png. 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.

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Middle

image49.pngimage49.png = 16image50.pngimage50.png

P7 = image51.pngimage51.png = 64image52.pngimage52.pngS7 = image53.pngimage53.png = 64image54.pngimage54.png

P10 = image55.pngimage55.png = 512image57.pngimage57.png

S10 = image58.pngimage58.png = 512image59.pngimage59.png

From a standard matrix image60.pngimage60.png, we can see that the difference between image20.pngimage20.png and image21.pngimage21.png, image22.pngimage22.png and image23.pngimage23.png is two. Therefore we could infer that it follows the pattern image03.pngimage03.png, and that image02.pngimage02.png in matrix P is two whereas image02.pngimage02.png 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 image03.pngimage03.png.

Steps 1 and 2 contain examples of these matrices for image02.pngimage02.png= 1, 2 and 3.

Consider other values of image02.pngimage02.png, and describe any pattern(s)

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Conclusion

image02.pngimage02.png, be it negative, rational, whole, will be coherent with the pattern. The values of image04.pngimage04.png 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 image04.pngimage04.png values were to be solely plugged into the pattern 2n-1image25.pngimage25.pngand not Mn , then similar to image02.pngimage02.png 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 .

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

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