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Matrix Binomials IA

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Introduction

Tenzin Zomkey                Maths SL Type 1

Maths Portfolio Standard Level

International Baccalaureate

Matrix Binomials

The main aim of this portfolio is to investigate the matrix binomials and observe and determine a general expression from the patterns that we obtain through the workings. Throughout the project, I shall be using solely matrices of 2 x 2 formations, and investigate the patterns I find.

1. To begin with, we consider the matrices X = image00.pngand Y =image01.png.

The values of these matrices, each raised to the power of 2, 3 and 4 are calculated, as shown below;

X2= image00.pngX image00.png= image43.pngY2 = image01.png x image01.png= image12.png

X3= image43.pngx image00.png= image02.png       and     Y3 = image12.pngx image01.png = image27.png

X4 = image02.pngx image00.png= image30.pngY4 = image27.pngx image31.png = image32.png

It can be observed that all the matrices calculated above are in the form of 2 X 2, they are all square matrices. The corresponding diagonal elements are also observed to be the same. Since the matrices of each nth power can be seen to be the value of 1 less than the nth term, the general expression for the matrix Xn in terms of n is -

Xn = image33.png

And the general expression for Yn is –

Yn = image34.png

...read more.

Middle

image48.pngB3= image49.png= image50.png

A4= image51.png= image52.pngB4= image53.png= image54.png

Note: A GCD calculator (TI 83) has been used throughout this portfolio to calculate the matrices and other calculations.

Observing the above calculations, we can detect a certain pattern in determining the values, which gives us the general expression of A and B in terms of n as;

An = image55.png               and                      Bn= image56.png

Proof:

Taking n to be 4, we substitute the values in the expression of An

A4= image57.png

A4 = image58.png

A4 = image59.png

A4 = image52.png

And since A4 = image52.pngis the correct value as calculated previously, this expression is proven true and consistent.

Likewise, to find the general expression of (A + B) n , the values of (A + B) raised to the powers 2,3 and 4 are calculated;

First, we find the value of (A + B)

(A + B)= image43.png+ image44.png= image60.png

(A + B) 2 = image03.png= image04.png

(A + B) 3 = image05.png= image06.png

(A + B) 4 = image07.png= image08.png

Observing the repeating patterns in the calculations above, we can deduce the general expression of (A+B) in terms of n to be;

(A + B) n = 2 n-1 image09.png

Proof:

Taking n as 3, and substituting it in the above expression –

(A + B) 3 = 2 3-1 image28.png

(A + B) 3 = 2 2image61.png

(A + B) 3 = 4 image11.png

(A + B) 3 = image06.png

...read more.

Conclusion

n = 2 n-1 image09.pngwhereby a=2, b= -2, and taking n=3, we calculate (A+B) raised to the third power –

(A+B)3 = 23-1 image10.png

                 = 22 image11.png

                  = 4 image11.png

                    = image06.png

So since we know from previous calculations that M3= image06.png, we can say that , M3= (A+B)3 .

Therefore, the general statement of Mnin terms of aX and bY is;

Mn= (aX+bY)n

Proof:

To check the validity of this general statement, we shall take different values for a, b and n. Suppose a= 3, b=4, and n= 2 –

M2= (3X+4Y)2

M2= image13.png

M2= image14.png

M2= image15.png

M2= image16.png

And since M = (A+B) = (aX+bY),

M = image17.png

M = image18.png

So then,

M2= image15.png= image16.png

Therefore, the statement is proven true and consistent with all values of a, b and n.

5.  Using the Algebraic method, the general statement is to be verified and explained again.

Taking the expression, Mn =2 n-1 image09.png, we find:

M = image19.png

M2= image20.png= 2image21.png

M3=  image22.png

 = 2 image23.png

= 2image24.png

= 2image25.png

=4image26.png

Proof:

Substituting the above with the initial values of a and b, we find M3 -

M3= 4 image26.png

M3 = 4 image28.png

M3= 4 image29.png

M3= 4image29.png

M3= 4image11.png

M3= image06.png

Therefore, since it has been shown earlier in our work that the value M3= image06.pngis true and correct, it shows that the general statement of Mnin terms of aX and bY is true.

...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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