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Math Portfolio

Extracts from this document...

Introduction

Shewit Aregawi Hagos

SL type 1 portfolio

Matrix Binomial

Submitted to: Ato Nestanet

By: Shewit Aregawi Hagos

Introduction

Matrices are rectangular tables of numbers or any algebraic quantities that can be added or multiplied in a specific arrangement. A matrix is a block of numbers that consists of columns and rows used to represent raw data, store information and to perform certain mathematical operations.

In this portfolio we are asked to generate different expressions from a series of matrices to the power of n and from this generate general statements that are appropriate to each question. Using a graphical display calculator all the basic mathematical calculation was made easier.

The first part of the question asks us to findimage00.pngimage00.png,image56.pngimage56.pngimage132.pngimage132.pngimage144.pngimage144.pnggiven that:

image152.png

image01.pngimage01.png

image08.pngimage08.png

image01.pngimage01.png

image30.pngimage30.png

and

image41.png

image44.pngimage44.png

image57.pngimage57.png

image68.pngimage68.png

From the above an obvious trend emerges whereby when we increase the power of the matrix there is an increase in the resulting matrix. For the X matrices I observed that when X2, the number in the resulting matrix are all 2’s; when X3, the numbers in the resulting matrix are all 4’s; when X 4, the numbers in the resulting matrix are all 8’s. A similar pattern has emerged in the Y

...read more.

Middle

image04.png

image05.png

Because image06.pngimage06.pngcan be expressed as image07.pngimage07.png then image09.pngimage09.pngcan be expressed like image10.pngimage10.pngwhich can be expressed in a matrix form, like it is shown below.

image11.pngimage11.pngimage12.pngimage12.png=image13.pngimage13.png

However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, A. In order to find the final expression for image14.pngimage14.png, we must multiply the general scalar valueimage15.pngimage15.png  the final general expression being:                image16.pngimage16.pngA

A similar technique can be used to determine the expression for image17.pngimage17.pngwhich is equal to bY thereforeimage18.pngimage18.png

When b= 1 then image19.pngimage19.png

image20.png

When b= 2 then image21.pngimage21.png

image22.png

Because image23.pngimage23.pngcan be expressed as image24.pngimage24.png then image25.pngimage25.pngcan be expressed like image26.pngimage26.pngwhich can be expressed in a matrix form, like it is shown below.

image27.pngimage27.pngimage28.pngimage28.png=image29.pngimage29.png

However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, B. In order to find the final expression for image14.pngimage14.png, we must multiply the general scalar valueimage31.pngimage31.png , the final general expression being:                image32.pngimage32.pngB

To find the expression forimage33.pngimage33.png a similar technique as before may be used; we know that A=aX and B=bXimage34.pngimage34.png. From this we can develop an expression forimage35.pngimage35.png:

image36.pngimage36.pngimage37.pngimage37.png

image38.png

image39.png

image40.pngimage40.png=image42.pngimage42.png

image36.pngimage36.pngimage43.pngimage43.png

The expression for image36.pngimage36.png is then image42.pngimage42.png

image45.png

image46.pngimage46.png

Example:

When a=-1

image47.pngimage47.png=image48.pngimage48.png

When b=2

image49.pngimage49.png=image50.pngimage50.png

When

...read more.

Conclusion

M2=A2+B2

image88.pngimage88.png= (aX) 2+(bY)2

image90.png=image91.png+image92.png

image90.png=image90.png

image93.png

To further illustrate that image87.pngimage87.png I have substituted integer values of a and b.

image94.pngimage94.png Let a= -1 and b= -1

image96.png

image97.png

Finally we are asked to come to a general statement for image98.pngimage98.pngin terms of aX and bX.   From all the calculation thus far we can deduce thatimage99.pngimage99.png therefore we can find the general statement by adding the expressions developed for image100.pngimage100.png:

image101.png

image102.pngimage102.png=image103.pngimage103.png

image104.pngimage104.png

To validate my general statement I must now substitute values for n, a and b:

image105.pngimage105.pnga=4 and b= 6

A=image106.pngimage106.pngimage108.pngimage108.png

image109.png

image110.png

Using my general statement I should be able to come up with the same value for A+B:

image14.pngimage14.png = aX image111.pngimage111.png                    and                   image112.pngimage112.png = bXimage113.pngimage113.png

image114.pngimage114.png=image115.pngimage115.png

image116.pngimage116.png, Proving that my general statement is indeed correct.

image117.pngimage117.pngn=3 a = 10 and   b = 3

image119.pngimage119.pngimage120.pngimage120.png

image121.png

image122.png

Again using my general statement I should be able to get the same value for A+B:

image14.pngimage14.png = aX image123.pngimage123.png                    and                   image112.pngimage112.png = bXimage125.pngimage125.png

image126.png

image122.pngimage122.png, once againproving that my general statement is indeed correct.

Limitations/Scope

  • As is one of the characteristics of matrices, fractions cannot be put as a power only integers can be powers. Thus using a non integer on the formula would not generate an appropriate answer.
  • The general formula also doesn’t work for negative number as matrices can’t have negative powers.
  • Another limitation is that A B and n must not equal each other.

...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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