• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

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=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.

                                     



Page |

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    TMRl��M6�Yc&#���#c�]���(��x7 �'���x��/��W�d�!�\6R�V'Q�&$@�%P5I�a�Xa�]^�5E� Ç¥8 )��x7 �'����"����bSH�a^'u8J2*�"H ��))nk,q,+��r9���js��o�v"1/4 ��� TSR�3/4�&"�D�1/4|� �U�����*�?�1/4 "���$E0L���31/4��t �� �0�?�| ,"�"�� ZUb(tm)A`'��+�1/4�۶n�z���K}���ZZZZ[[��}��mÛ¶m�Ν"v�*f|�Xx<-�B�X-�`-�1/4�H 'f!)�{�'�x�� L$R/_x�...3/43/43/4��^����"�z�(c)={�=z�""�'#Y:�c�BCa�"��c.��"��-�k^'/ ���...��(c)(c)A�\.�D2 ���W," |�}�����O1/4b���XM�=��4[�B>h�(r)�O��$@�$0 I!u��ޱc��-'QWWw��7�}�����x{����Z����b$.6f��qW���>�(c)� ���/�"" ("��$ų�>���?��R�z�-_~�n��yç¨ï¿½ �x'��bK�z�%W�S�"�_��t�mÌrx�OY P�YH��_� � �� ��^���;�x�(tm)g���ÆGGG�Q lÙ²E*CR@�H��c��F�'�Ö��J�:� (tm)|kH��H XI'�fo1/2�� -x@�V�[���f�/�Eqxx���"w���...-MW3/4��#i>��0�T U...�J���p���B �1/4�8�X�f D�^-8p`Ñ¢E��~;�'N�:�-�-���'���{/G{{�w3/4��C�-�� �u�z0k�el�A(c)�d�K�����|�e|K$P-S�n�� �`d�P��x<-�D���z(c)<��h��_�ܵ��"J�;$�<gO����'�5$@U!0/�N@�C��(r);�_...5-,�`�(tm)�!�(r) �m���S�%���$�b�e�" &��0�L�"��&bU�<�(+qÞfI@'�7� �l= �@�+h7'�x-��$@e$ I���0' ���b|�F��}�'{�8nÞH`6!)/h�Q��OB��ceôºï¿½ï¿½%(#EH �/J3"��"-��E&�>o^����$@�P��' s<'�� ��)D� c� �W' "��R$b�`A!�T�!m�>�l��% I�>v�`2"ò¤ª@�1/4�iܸ"Ø�: �2P����}a��(tm)��E�jg��)����$0J'�>g���P�jC����3/4���f3(tm)1/4-�F@A'BH�"T2���V��6�_X����$@�

  2. Math Portfolio: trigonometry investigation (circle trig)

    x sinx cosx 0 0 1 1 0 - 0 -1 -1 0 0 1 y=3cos (?) y= (-3)cos? y=3cos (-?) The maxima of this graph y=3cos? is 3 and the miima is -3. The amplitude in the y=3cos? is resulting to 3.

  1. Maths SL Portfolio - Parallels and Parallelograms

    = sum of all integers from 1 to 9 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 p = (m2 - m) � 2 p = 102 - 10 � 2 = 90 � 2 = 45 We

  2. Math IA - Matrix Binomials

    in order to yield the product (X+Y)n. In order to find the final expression for (X+Y)n, we must multiply the general scalar value 2n-1 by matrix (X+Y): , which can also be written as . In order to test the validity of this expression, we can employ it to find (X+Y)5.

  1. Stellar Numbers math portfolio

    general statement that represents the nth triangular number in terms of n is: or in factored from. Restrictions please. The second way that this general statement could be derived is with technology. It has already been established that because the second differences are constant and not zero this, the general statement is to the second degree.

  2. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    This would be attributed to Wessig?s surprise performance at the 1980 event; Wessig covertly used a new unique technique not used before which contributed to his breaking the world record by a wide margin. Given the new data expansion, it would seem appropriate to modify the model so that it fits much more with the extra data points.

  1. Math SL Fish Production IA

    Table 11: This shows the total mass of fish, in thousands of tonnes from fish farms. Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 Total Mass 1.4 1.5 1.7 2.0 2.2 2.7 3.1 3.3 4.1 Year 1989 1990 1991 1992 1993 1994 1995 1996 1997 Total Mass 4.4

  2. Mathematic SL IA -Gold medal height (scored 16 out of 20)

    same points with Figure 2, but this seems little different with what shape figure 2 has. This is because figure 2 has y-axis which starts from 195. But in figure 7, it has origin (0, 0). Therefore we can see the same graph in different appearance.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work