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math portfolio type 1

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Introduction

IB Mathematics SL

Portfolio Type I

Matrix Powers

Done by: Bassam Al-Nawaiseh

IB II

  • Introduction:

Matrices are rectangular tables of numbers or any algebraic quantities that can be added or multiplied in a specific arrangement. A matrix is a block of numbers that consists of columns and rows used to represent raw data, store information and to perform certain mathematical operations. The aim of this portfolio is to find general formulas for matrices in the form    .        image00.pngimage01.pngof

Each set of matrices will have a trend in which a general formula for each example is deduced.

  • Method 1:

Consider the matrix M =image17.pngimage18.png when k = 1.

Table 1: Represents the trend in matrix M = image17.pngimage18.pngas n is changed in each trial.                                

Power

Matrix

n = 1

image17.pngimage18.png

n = 2

image96.pngimage101.png

n = 3

image02.pngimage13.png

n = 4

image16.pngimage25.png

n = 5

image34.pngimage43.png

n = 10

image50.pngimage51.png

n = 20

image54.pngimage61.png

Matrix M is a 2 x 2 square matrix which have an identity. As n changes the zero patterns is not affected while the 2 is affected. 2n is raised to the power of n. When n =1, 21 = 2, when n = 2, 2² = 4 and when n = 3, 2³ = 8 and so on. So as a conclusion, Mn = image19.pngimage20.png

  • Method 2

Consider the matrices P = image21.pngimage22.png

...read more.

Middle

image10.png1 = 20image09.pngimage10.png = image09.pngimage10.png

n = 2

image09.pngimage10.png2 = 21 image59.pngimage60.png= image55.pngimage56.png

n = 3

image09.pngimage10.png3= 22image97.pngimage98.png = image99.pngimage100.png

n = 4

image09.pngimage10.png4= 23image14.pngimage15.png= image11.pngimage12.png

n = 5

image09.pngimage10.png5= 24image102.pngimage103.png= image03.pngimage04.png

  As for two consecutive matrices, the trend is found so as the power n increases by a factor of one. The scalar is doubled in each trial and then a certain factor is added to the elements of matrix S. taking n = 2 as an example, the scalar is found by doubling the power n = 1 (1x2=2). The factor of addition is determined by multiplying the difference in x and y inside the matrix with a power less with one by 3. In matrix n = 1, the difference in the elements is 2 (4-2=2), the answer is multiplied by 3 (2x3=6). Finally 6 is added to the matrix of n = 2. The general formula for this trend is Sn = 2 image05.pngimage06.png where n = 1. Notice that this formula needs two consecutive matrix powers in order to be applied. Another general formula can be derived for this sequence. As the power n changes, the power in which the scalar is raised will change also.

Sn = 2n-1 image07.pngimage08.png where n is an integer. To check the validity of this formula n = 4 is used as an example.

Using GDC: image09.pngimage10.png4 =image11.pngimage12.png

Using the General Formula:image09.pngimage10.png4=23image07.pngimage08.png=8image14.pngimage15.png=image11.pngimage12.png.

  • Method 3:
  1. Consider the matrices in the form Q = image01.pngimage00.png.

Table 3: Represents the trend in matrix Q as k is increased by one in each trial.

Power

Matrix

General Formula

k = 1

M = image17.pngimage18.png

Mn = 2n-1 image19.pngimage20.png

k = 2

P = image21.pngimage22.png

Pn = 2n-1 image23.pngimage24.png

k = 3

S = image09.pngimage10.png

Sn = 2n-1image07.pngimage08.png

k = 4

D = image26.pngimage27.png

Dn = 2n-1image28.pngimage29.png

k = 5

F = image30.pngimage31.png

Fn = 2n-1image32.pngimage33.png

k = 6

N = image35.pngimage36.png

Sn = 2n-1image37.pngimage38.png

...read more.

Conclusion

image01.pngimage00.pngn, so matrix B = image52.pngimage53.png. From rule A and the above examples, the following formula is deduced: Bn = 2n-1image39.pngimage40.png

Using GDC: image52.pngimage53.png2 = image55.pngimage56.png

Using Rule A: image52.pngimage53.png2 = 22-1 image57.pngimage58.png = 2 image59.pngimage60.png = image55.pngimage56.png

As a conclusion, rule A also can be applied when k is a negative value.

  1. Fraction Values:

Matrix L represents the matrix where k is a fraction value raised to the power of n, k = 3/2 and n = 3 in a matrix of the form image01.pngimage00.pngn, so matrix L =image62.pngimage63.png. From the above examples, Ln = 2n-1image39.pngimage40.png.

Using GDC:  image62.pngimage63.png3 = image64.pngimage65.png

Using Rule A: image62.pngimage63.png3 = 23-1 image66.pngimage67.png= 4 image68.pngimage69.png = image64.pngimage65.png.

From the above results, it is shown that rule A is valid even for fraction numbers.

  1. Irrational Values:

Matrix U represents the matrix where k is an irrational number which is raised to the power of n, k = image70.png and n = 2 in a matrix of the form

image01.pngimage00.pngn, so matrix L = image71.pngimage72.png. From the above examples,

Un = 2n-1image39.pngimage40.png.

Using GDC: image71.pngimage72.png2 = image73.pngimage74.png.

Using Rule A: image71.pngimage72.png2 = 22-1 image75.pngimage76.png =

2 image77.pngimage78.png = image73.pngimage74.png

From those final results, we can prove that this statement, Rule A, can also be applied to irrational integers.

...read more.

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

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