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Portfolio SL - Matrix

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Introduction

Math Portfolio SL

MATRIX BINOMIALS

NONAME STUDENT

3/17/2009


MATRIX BINOMIALS

In this portfolio I will try to find a general statement and patterns for given matrix binomials exercises. For data processing I will use TI-83 Plus (Sliver Edition) graphing calculator. I will use my knowledge from patterns and matrix  in order to find suitable formulas.

X=image00.pngimage00.png, Y=image37.pngimage37.png.

Find X2, X3,X4;Y2, Y3,Y4.

image84.pngimage89.png

X2=image94.pngimage94.png= image103.pngimage103.pngX3=image01.pngimage01.pngX4=image02.pngimage02.png
Y2=image17.pngimage17.pngY3=image14.pngimage14.pngY4=image28.pngimage28.png

Find expressions for Xn , Yn, (X+Y)n

  • The entries double for every higher power of X, i.e.:

X2= 2X= 21X
X
3= 4X= 22X
X
4= 8X = 23X     follows
Xn=2n-1X 

I will test this formula with a random number:                                    X10=image38.pngimage38.pngX10= 210-1image46.pngimage46.png= image38.pngimage38.png 
image59.png

X19= 219-1image46.pngimage46.png= image71.pngimage71.png

Xn=2n-1Xformula valid for all natural numbers N;image79.png = {1,2,3,...} as it is shown in examples.

  • The entries double for every higher power of Y, i.e.:

Y2= 2Y
Y
3= 4Y= 22Y
Y
4= 8Y = 23Y

...read more.

Middle

A3=image05.pngimage05.png3= image06.pngimage06.pngA3= image03.pngimage03.png3= image07.pngimage07.png
A4= image05.pngimage05.png4=image08.pngimage08.pngA4= image03.pngimage03.png4=image09.pngimage09.png

image10.pngimage11.png

b=3                                                         b=4
B=3Yimage12.pngimage12.pngB= image13.pngimage13.png                                   B=4Yimage12.pngimage12.pngB= image14.pngimage14.png
B
2=image13.pngimage13.png2= image15.pngimage15.pngB2= image14.pngimage14.png2= image16.pngimage16.png

B3=image13.pngimage13.png3= image18.pngimage18.pngB3= image14.pngimage14.png3= image19.pngimage19.png
B4=image13.pngimage13.png4= image20.pngimage20.pngB4= image14.pngimage14.png4= image21.pngimage21.png

Find expressions for An , Bn, (A+B)n

  • As I’m raising powers by one, the entries increase double the constant is.
    For example:

A= 6Ximage12.pngimage12.png A=image22.pngimage22.png     
A
2= image23.pngimage23.png              A3=image24.pngimage24.pngA4= image25.pngimage25.png
As it is visible, every entry increases by 12, i.e. double the constant
a is.

From this pattern I derived formula that:
An= anXn

Testing with random numbers:
A2=62X2
A2=36image05.pngimage05.png= image23.pngimage23.png

A
=10Ximage12.pngimage12.png A=image26.pngimage26.pngA3=103X3                    
A3=image26.pngimage26.png3= image27.pngimage27.pngA3= image27.pngimage27.png

The same pattern goes for
B=bY.

image29.png
B=-3Y
B2=image30.pngimage30.png2= image31.pngimage31.png

B3=image30.pngimage30.png3= image32.pngimage32.png                     
B4=image30.pngimage30.png4= image33.pngimage33.png          The entries increase by 6, i.e. the double b is.    

image34.png
B
=image35.pngimage35.pngY
B≈
image36.pngimage36.png
B
2image36.pngimage36.png2image39.pngimage39.png
The entries increase by app. 6.26, i.

...read more.

Conclusion

+Bk)(A+B)= AkA + BkB +AkB + BkA

We showed that AB=BA=0, therefore AkB + BkA=0

It follows that
AkA + BkB +AkB + BkA= AkA + BkB  
A
kA + BkB= Ak+1 + Bk+1
Mk+1= Ak+1 + Bk+1

By doing and deriving formulas (especially in the second exercise) I found that the general formula is:
Mn=anXn+ bnYn
                                                         
 Mn=(aX+ bY)n


a

b

n

5

2

 2

   
M= image66.pngimage66.pngM= image67.pngimage67.png

 M2= image68.pngimage68.png
image67.pngimage67.png2= image69.pngimage69.png2
image67.pngimage67.png2= image70.pngimage70.pngn

image67.pngimage67.png2= image67.pngimage67.png2
Using GDC:
M2= image72.pngimage72.png

a

b

n

4

-1.5

 5

   
M= image66.pngimage66.pngM= image73.pngimage73.png
image73.pngimage73.png5=image74.pngimage74.png5
image73.pngimage73.png5= image73.pngimage73.png5
image75.png

M5= image76.pngimage76.png

In the conclusion I can say that I found the scope of the statement.
|A|= ab-cd      
|X|
= image77.pngimage77.png|Y|= image78.pngimage78.png                             
|X|= 1-1= 0                              |Y|= 1-1= 0                            

Since the starting matrices have determinant 0 and thus they don’t have inverse, in my conclusion I can say that n€image79.png. Also I showed in examples above that constants a and b can be rational and irrational numbers (image80.pngimage80.png, therefore a, b€R. I would limit my general statement on the set of real numbers this is the only set of number that we learnt in matrix unit.

                                     



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...read more.

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