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Matrix Binomials Portfolio

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Introduction

Math SL Matrix Binomials Portfolio

        This portfolio will investigate the properties of matrix binomials in order to determine a general statement for Mn where n is a real number and an integer, and M is the 2image00.pngimage00.png matrix image28.pngimage28.png

Let:

X = image48.pngimage48.png                                 Y= image59.pngimage59.png

X2 = image70.pngimage70.png = image01.pngimage01.png                           X3 = image12.pngimage12.png = image24.pngimage24.png

                                       X4 = image34.pngimage34.png

Y2 = image37.pngimage37.png        Y3 = image38.pngimage38.png

                                       Y4 = image39.pngimage39.png

(X+Y) = image40.pngimage40.png

(X+Y)2 = image41.pngimage41.pngimage42.pngimage42.png                    (X+Y)3 = image43.pngimage43.png

                                     (X+Y)4 = image44.pngimage44.png

Expressions for Xn, Yn and (X+Y)n

Xn = image45.pngimage45.png            Yn = image46.pngimage46.png       (X+Y)n = image47.pngimage47.png

n > 0,  

let: W = any 2x2 matrix, image49.pngimage49.png

W-n = image50.pngimage50.png It is not possible to

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Middle

image57.pngimage58.pngimage58.png

B3 = image60.pngimage60.png

B4 =  image61.pngimage61.png

Therefore:

(A+B) = image62.pngimage62.png

(A+B)2 = image63.pngimage63.png

image64.pngimage64.png

image65.pngimage65.png

(A+B)3 = image66.pngimage66.png

image67.pngimage67.png

(A+B)4 = image68.pngimage68.png

image69.pngimage69.png

Expressions for An, Bn and (A+B)n

An = image71.pngimage71.png           Bn = image72.pngimage72.png

                         (A+B)n = image73.pngimage73.png

n > 0,  

let: W = any 2image74.pngimage74.png2 matrix, image49.pngimage49.png

W-n = image50.pngimage50.png It is not possible to divide an integer by a matrix, n < 0 does not exist

n≠0

For any matrix where n=0 Wn = I    W0 = image51.pngimage51.png

Let: M = image28.pngimage28.png, M = A+B and M2 = A2+B2

A = aX = image13.pngimage13.png

...read more.

Conclusion

n must also be greater than zero. It cannot equal zero because any matrix raised to the power of zero would equal I the identity matrix, and it cannot be less than zero, because that would be the same as one divided by the matrix raised to the power of n. Since it is not possible to divide an integer by a matrix, it is not possible to raise any matrix to a negative exponent and this n must be greater than zero. Therefore the expressions determined in this portfolio are valid for 2image36.pngimage36.pngmatrices, where the value of n is greater than zero

...read more.

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

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