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# Portfolio SL - Matrix

Extracts from this document...

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=, Y=.

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

X2== X3=X4=
Y2=Y3=Y4=

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=X10= 210-1=

X19= 219-1=

Xn=2n-1Xformula valid for all natural numbers N; = {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

Middle

A3=3= A3= 3=
A4= 4=A4= 4=

b=3                                                         b=4
B=3YB=                                    B=4YB=
B
2=2= B2= 2=

B3=3= B3= 3=
B4=4= B4= 4=

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= 6X A=
A
2=               A3=A4=
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=36=

A
=10X A=A3=103X3
A3=3= A3=

The same pattern goes for
B=bY.

B=-3Y
B2=2=

B3=3=
B4=4=           The entries increase by 6, i.e. the double b is.

B
=Y
B≈

B
22
The entries increase by app. 6.26, i.

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= M=

M2=
2= 2
2= n

2= 2
Using GDC:
M2=

 a b n 4 -1.5 5

M= M=
5=5
5= 5

M5=

In the conclusion I can say that I found the scope of the statement.
|A|= ab-cd
|X|
= |Y|=
|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€. Also I showed in examples above that constants a and b can be rational and irrational numbers (, 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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