- Level: International Baccalaureate
- Subject: Maths
- Word count: 840
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=, 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
X3= 4X= 22X
X4= 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
Y3= 4Y= 22Y
Y4= 8Y = 23Y
Middle
A3=3= A3= 3=
A4= 4=A4= 4=
b=3 b=4
B=3YB= B=4YB=
B2=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=
A2= 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≈
B2≈ 2≈
The entries increase by app. 6.26, i.
Conclusion
We showed that AB=BA=0, therefore AkB + BkA=0
It follows that
AkA + BkB +AkB + BkA= AkA + BkB
AkA + 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.
Page |
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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