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Matrix power. The internal assessment will focused on observing patterns of matrix powers which will be the main key to find the general expression of matrix powers.

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Introduction

Internal Assessment- Matrix Powers

        The internal assessment will focused on observing patterns of matrix powers which will be the main key to find the general expression of matrix powers. image00.png

image04.pngimage01.png

1)

Explanation: When matrix M is powered by 2 it gives a result ofimage21.pngfrom GDC, when M is powered by 3 it gives a result of image31.pngfrom GDC and when M is powered by 4 it gives a result of image41.pngfrom GDC. The pattern shown is that every time M is powered by a number after its preceding number it is multiply by 2. For instance, 4 shown in matriximage48.png, 8 shown in matriximage58.png, and 16 shown in matrix image59.png. Thus, these results make a general expression thatimage47.png. This formula is proven correct because whenimage02.png

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Middle

image15.pngsuch as 7 when n = 3, 9 when n = 4 and then 31 when n = 5. Thus, these results make a general expression thatimage16.png. This formula is proven correct because whenimage17.png from GDC and this formula also works whenimage18.pngfrom GDC.image04.png

image20.pngimage19.png

Explanation: When matrix S is powered by 3 it gives a result of image22.pngfrom GDC, when matrix S is powered by 4 it gives a result ofimage23.pngfrom GDC, and when matrix S is powered by 5 it gives a result of image24.pngfrom GDC. The pattern shown is that results have a common factor of image10.pngsuch as 4 shown in matriximage25.png, 8 shown in matriximage26.png, and then16 shown in matriximage27.png. Once the resulting matrix is factor out, the

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Conclusion

image40.png is equivalent to result matriximage42.png, where as the results of image43.pngis equivalent to result matrix image44.png and the results of image45.png is equivalent to result matriximage46.png. The pattern also shows that whenimage40.pngthe matrix formula isimage47.png, when image43.pngthe matrix formula isimage47.png, and whenimage45.png the matrix formula isimage32.png. This shows that any number representing image49.pngcorresponds toimage50.pngandimage51.png. Thus, these results give a general expression thatimage52.png. For instance, if image53.pngand image54.png then image55.png =   image56.pngfrom GDC.

4)

Explanation: After a further investigation with further values of k and n, the results shows that the limitation for n is that image57.pngall negative numbers and non-integers value because when a matrix is powered by negative numbers or non-integers value the calculator gives a math error text. In addition there’s no limitation for k.  

5)

In conclusion,

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