matrix binomilas-portfolio

IB PORTFOLIO-  MATRIX BINOMIALS

MATH SL

JAIME CASTRO A.

10-2

PRESENTED TO:

CLAUDIA GOMEZ

ANGLO COLOMBIAN SCHOOL

MATH DEPARTMENT

BOGOTA, MARCH 2009

INTRODUCTION

This project is about 2x2 matrices or matrix binomials; the aim of this work is to demonstrate a good and clear understanding of matrices and the operations that can be done with them.

At the end of the project we should have been able to generate different expressions for some different matrices powered by an n integer as well as a general statement to calculate the addition of specific matrices powered to an n integer.

To have a correct development of this piece of work it is essential that each question is solved in order since the result of a question will be necessary for the development of the following one.

Since there are many basic calculations that aren’t really important we will use a GDC (graphic display calculator) in order to speed up the development of the project.

QUESTIONS

1)

Let  and. Calcúlate , ; , .

With the help of a GDC (Graphic display calculator) the calculations for the matrices were done and registered below.

2)

By considering different integer powers of X and Y find an expression for,  and.

We have already calculated the resulting matrices for three different integer powers of X and Y in the first question, however to find an expression for,  and and be sure that it is true, we are also going to calculate the resulting matrices of using 5,6 and 7 as powers of X and Y.

The,;, shown below were calculated by using a GDC.

By analyzing the resulting matrices of different powers of X and Y we can develop an expression for and  which are shown below.

Now to generate an expression for  we must first develop the matrix resulting from X+Y so we can then use different powers for (X+Y) and hence generate the expression for

X + Y=  + =

(X + Y)=

By analyzing the resulting matrices of different powers of (X+Y) we can develop an expression for which is shown below.

3)

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