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# matrix binomilas-portfolio

Extracts from this document...

Introduction

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.

Middle

.

In the case of a being 8 ,

While in the case of a being 3 ,

So if can be expressed like  then can be expressed like which can be expressed in a matrix form, like it is shown below.

=

=

To find an expression for  we can do the same process done to generate the expression for  , so we can star by stating that isby definition, so should be equal to  which can also be expressed as), we can check this by diving each element of both of the calculated previously by their respective values of and the result should be equal to.

In the case of b being 2 ,

While in the case of b being 5 ,

So if  can be expressed like  then can be expressed like which can be expressed in a matrix form, like it is shown below.

=

=

The expression for  can be found in the same way that we found the expressions for and , we know that by definition A is aX and isso  is the same as. Starting from this we can develop an expression for  as shown below.

=

The expression for  is then

4)

Now consider

Show that

We know that matrix = so know we just need to add matrix A to matrix B and see if the resulting matrix is the same as

Conclusion

We have then

=

=

Now if we use our general statement it should give use the same when we replace the values of a, b and n with the ones used to develop

==

==

Once again the statement worked and therefore I can consider it a general statement.

CONCLUSION

Nb b

• The general statement developed did worked correctly and was checked two times with different values for a, b and n.
• The relationship between the matrix with the matrices and  was found and it was expressed in matrix form in question 4.
• We were able to generate expressions for specific matrices powered by an unknown n integer and having developed those expressions we were able to generate a general statement.

EVALUATION

In my opinion the project was developed in a good and coherent manner, there weren’t any serious problems or difficulties during the processes and the results obtained were satisfactory, since I was able to generate the general statement and the other expressions for the different matrices powered by an n integer.

During the process I used all my knowledge about matrices which helped me to develop a good understanding of the questions in order to obtain the different answer.

However I think that the information in some parts is not presented in an organized way, this could have been improved with the use of tables to record the information in a more suitable place.

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

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