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matrix binomilas-portfolio

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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 image00.pngimage00.png andimage47.pngimage47.png. Calcúlate image11.pngimage11.png, image129.pngimage129.png; image139.pngimage139.png, image01.pngimage01.png.

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

image13.png

image18.png

image23.png

2)

By considering different integer powers of X and Y find an expression forimage27.pngimage27.png, image37.pngimage37.png andimage45.pngimage45.png.

...read more.

Middle

.

In the case of a being 8 , image12.pngimage12.png

image14.png

While in the case of a being 3 , image15.pngimage15.png

image16.png

So if image06.pngimage06.pngcan be expressed like image17.pngimage17.png then image04.pngimage04.pngcan be expressed like image19.pngimage19.pngwhich can be expressed in a matrix form, like it is shown below.

image20.pngimage20.pngimage21.pngimage21.png=image22.pngimage22.png

image04.pngimage04.png=image24.pngimage24.png

To find an expression for image05.pngimage05.png we can do the same process done to generate the expression for image04.pngimage04.png , so we can star by stating that image25.pngimage25.pngisimage26.pngimage26.pngby definition, so image28.pngimage28.pngshould be equal to image29.pngimage29.png which can also be expressed asimage30.pngimage30.png), we can check this by diving each element of both of theimage28.pngimage28.png calculated previously by their respective values of image31.pngimage31.pngand the result should be equal toimage32.pngimage32.png.

In the case of b being 2 , image33.pngimage33.png

image34.png

While in the case of b being 5 , image35.pngimage35.png

image36.png

So if image28.pngimage28.png can be expressed like image38.pngimage38.png then image05.pngimage05.pngcan be expressed like image39.pngimage39.pngwhich can be expressed in a matrix form, like it is shown below.

image40.pngimage40.pngimage41.pngimage41.png=image42.pngimage42.png

image05.pngimage05.png=image43.pngimage43.png

The expression for image44.pngimage44.png can be found in the same way that we found the expressions for image04.pngimage04.pngand image05.pngimage05.png, we know that by definition A is aX and image25.pngimage25.pngisimage26.pngimage26.pngso image44.pngimage44.png is the same asimage46.pngimage46.png. Starting from this we can develop an expression for image44.pngimage44.png as shown below.

image48.pngimage48.pngimage49.pngimage49.png

image50.png

image51.png

image52.pngimage52.png=image54.pngimage54.png

image48.pngimage48.pngimage55.pngimage55.png

The expression for image48.pngimage48.png is then image54.pngimage54.png

4)

Now consider image56.pngimage56.png

Show that image57.pngimage57.pngimage58.pngimage58.png

We know that matrix image59.pngimage59.png=image60.pngimage60.png so know we just need to add matrix A to matrix B and see if the resulting matrix is the same as image59.pngimage59.png

...read more.

Conclusion

We have then

image105.png

image106.pngimage106.png=image108.pngimage108.png

image106.pngimage106.png=image109.pngimage109.png

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 developimage110.pngimage110.png

image111.pngimage111.png=image113.pngimage113.png=image114.pngimage114.png

image111.pngimage111.png=image115.pngimage115.png=image116.pngimage116.png

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 image83.pngimage83.pngwith the matrices image118.pngimage118.pngand image119.pngimage119.png 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.

...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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