- Level: International Baccalaureate
- Subject: Maths
- Word count: 1175
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 . of
Each set of matrices will have a trend in which a general formula for each example is deduced.
- Method 1:
Consider the matrix M = when k = 1.
Table 1: Represents the trend in matrix M = as n is changed in each trial.
Power | Matrix |
n = 1 | |
n = 2 | |
n = 3 | |
n = 4 | |
n = 5 | |
n = 10 | |
n = 20 |
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 =
- Method 2
Consider the matrices P =
Middle
n = 2
2 = 21 =
n = 3
3= 22 =
n = 4
4= 23=
n = 5
5= 24=
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 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 where n is an integer. To check the validity of this formula n = 4 is used as an example.
Using GDC: 4 =
Using the General Formula:4=23=8=.
- Method 3:
- Consider the matrices in the form Q = .
Table 3: Represents the trend in matrix Q as k is increased by one in each trial.
Power | Matrix | General Formula |
k = 1 | M = | Mn = 2n-1 |
k = 2 | P = | Pn = 2n-1 |
k = 3 | S = | Sn = 2n-1 |
k = 4 | D = | Dn = 2n-1 |
k = 5 | F = | Fn = 2n-1 |
k = 6 | N = | Sn = 2n-1 |
Conclusion
Using GDC: 2 =
Using Rule A: 2 = 22-1 = 2 =
As a conclusion, rule A also can be applied when k is a negative value.
- 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 n, so matrix L =. From the above examples, Ln = 2n-1.
Using GDC: 3 =
Using Rule A: 3 = 23-1 = 4 = .
From the above results, it is shown that rule A is valid even for fraction numbers.
- Irrational Values:
Matrix U represents the matrix where k is an irrational number which is raised to the power of n, k = and n = 2 in a matrix of the form
n, so matrix L = . From the above examples,
Un = 2n-1.
Using GDC: 2 = .
Using Rule A: 2 = 22-1 =
2 =
From those final results, we can prove that this statement, Rule A, can also be applied to irrational integers.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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