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I.B. Maths portfolio type 1 Matrices

Extracts from this document...

Introduction

IB MATHS SL                PANJI WICAKSONO

MATHS PORTOFOLIO SL TYPE 1: MATRIX POWER

CANDIDATES NAME: PANJI WICAKSONO

IB NOVEMBER 2008

INTRODUCTION:

Matrices are an array of numeric or algebraic quantities subject to mathematical operations that can be multiplied, subtracted or added. The numbers can be arranged in a rectangular array of numbers set out in rows and columns. The numbers that are inside the matrices are called entries. Matrices are useful to keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. In a more complex use, it can be helpful in encrypting numerical data and also in computer graphics.

In this portfolio, the task that was given was to deduce a general formula or pattern of the given matrices and also to determine the limitation of the general formula that has been found. This task will be done using the help of a GDC calculator (GDC-TI 83 plus) and the knowledge that have been attained during the mathematic SL course.

QUESTION 1:

Consider the matrix

image00.png

Calculate Mnfor n = 2, 3, 4, 5, 10, 20, 50

Describe in words any pattern you observe.

Use this pattern to find a general expression for the matrix Mn in terms of n

Counting for n= 2 manually, it gives:

M2 = M.M =image01.pngimage02.pngimage01.png = image102.png

For n= 3
M3 = M2.M = image102.pngimage02.pngimage01.png = image127.png

For n = 4

M4 = M3.M =image138.pngimage02.pngimage13.png = image24.png

For n = 5

M5image33.png= M4.M = image24.pngimage02.pngimage13.png = image53.png

For the larger values of n, the GDC was used to help ease the task by first pressing MATRIXmenu EDIT option, and then chose matrix 1:[A]

...read more.

Middle

; S2= image119.png = image120.png = 2image45.png

Calculate Pn and Sn for other values of n and describe any pattern(s) you observe.

Pn and Sn was calculated using the GDC for values such as n = 3, 4, 5, 6, 7, 8, 9, 10

I started using the GDC by inserting matrices P and S to the memory slot. For P the matrix slot that was used was [B] and for S I used matrix slot [C]. I specifically entered those matrices in a 2 x 2 dimension to suit the purpose. After storing the matrix in the memory I accessed it using the MATRIX, NAMES submenu and raise it to the value of n that I was calculating.

The results I got by calculating P in the GDC was

P3= image121.png = image122.png

= 4image123.png

P7= image124.png = image125.png

= 64image126.png

P4= image128.png = image129.png

= 8image130.png

P8= image131.png = image132.png

= 128image133.png

P5= image134.png = image135.png

= 16image136.png

P9= image137.png = image139.png

= 256image140.png

P6= image141.png = image142.png

= 32image143.png

P10= image141.png = image144.png

= 512image145.png

For the second part I will calculate Sn using n as 3, 4, 5, 6, 7, 8, 9, and 10.

S3=image146.png = image147.png

= 4image03.png

S4=image04.png = image05.png

= 8image06.png

S5=image07.png = image08.png

= 16image09.png

S6= image10.png = image11.png

= 32image12.png

S7= image14.png = image15.png

= 64image16.png

S8= image17.png = image18.png

= 128image19.png

S9= image20.png = image21.png

= 256image22.png

S10= image23.png = image25.png

= 512image26.png

By calculating the different values of n for Pn and Sn a specific pattern could be observed, which is:

Xn= 2(n-1)image27.pngimage28.png

X in this general expression corresponds to the matrix P or S, and R inside the matrix is 2 when P is used, and it is 3 when S is used. I will now try to recheck whether the general formula by using the general formula that I observed and comparing it to the results calculated using the GDC.

P5= 2(5-1)image29.pngimage30.png= 24image31.png= 16image31.png

S5= 2(5-1)image29.pngimage32.png= 24image09.png= 16image09.png

The general formula gave the same answer as to the result gathered using the GDC. That means the general formula is acceptable.

QUESTION 3

...read more.

Conclusion

k = 1

n = 0

Solving Manually:

X0 = image47.png

Solving Technologically:

X0 =image48.png=image49.png

k = 1

n = 5

Solving Manually:

X0 = image50.png

Solving Technologically:

X5 =image51.png=image52.png

k = 1

n = image54.png2

Solving Manually:

Ximage54.png2 = image55.png

Solving Technologically:

Ximage54.png2 =image56.png= ERR: DOMAIN

k = 1

n =image57.png

Solving Manually:

Ximage58.png=image59.png

Solving Technologically:

Ximage58.png= image60.png= ERR: DOMAIN

\k =1

n = image62.png

Solving Manually:

Ximage63.png=image64.png

Solving Technologically:

Ximage63.png= image65.png= ERR: DOMAIN

k = 1

n = 3.75

Solving Manually:

X3.75 = image66.png

Solving Technologically:

X3.75 = image67.png= ERR: DOMAIN

k = 1

n = - 3.75

Solving Manually:

X-3.75 = image68.png

Solving Technologically:

X-3.75 = image69.png= ERR: DOMAIN

k = 1

n = image70.png

Solving Manually:

Ximage72.png= image73.png

Solving Technologically:

Ximage72.png = image74.png= ERR: DOMAIN

k = 1

n = image75.png

Solving Manually:

Ximage76.png= image77.png

Solving Technologically:

Ximage76.png=image78.png= ERR: DOMAIN

k = 1

n = 255

Solving Manually:

X255 = image79.png

image80.png

Solving Technologically:

X255 =image81.png= image82.png

k = 1

n = 256

Solving Manually:

X256 = image83.png

image84.png

Solving Technologically:

X255 =image85.png= ERR: DOMAIN

From the results above, it is evident that some types of numbers are not suitable when being calculated by the GDC. Those numbers include, work for negative whole numbers, fractions, negative fractions, decimals, negative decimals, surds, irrational numbers and numbers greater than 255 as proved using the GDC above.

However, the general formula does work when the value of n is a positive integer, a shown above using the value n = 0, 5 and 255. Thus it can be concluded that the general equation only works when the value of n is a positive integer.

On the other hand, any value of K is acceptable and can be used in this general formula as long as the entries inside the matrix has a difference of 2, which it follows the matrix formula of k+1 and k-1. If the matrix has a difference of 2 (regardless of it being negative, positive, fractions, decimals, irrational, surds etc.) the general formula would work.

...read more.

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