- Level: International Baccalaureate
- Subject: Maths
- Word count: 1041
Math Portfolio
Extracts from this document...
Introduction
Shewit Aregawi Hagos
SL type 1 portfolio
Matrix Binomial
Submitted to: Ato Nestanet
By: Shewit Aregawi Hagos
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.
In this portfolio we are asked to generate different expressions from a series of matrices to the power of n and from this generate general statements that are appropriate to each question. Using a graphical display calculator all the basic mathematical calculation was made easier.
The first part of the question asks us to find,given that:
and
From the above an obvious trend emerges whereby when we increase the power of the matrix there is an increase in the resulting matrix. For the X matrices I observed that when X2, the number in the resulting matrix are all 2’s; when X3, the numbers in the resulting matrix are all 4’s; when X 4, the numbers in the resulting matrix are all 8’s. A similar pattern has emerged in the Y
Middle
Because can be expressed as then can be expressed like which can be expressed in a matrix form, like it is shown below.
=
However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, A. In order to find the final expression for , we must multiply the general scalar value the final general expression being: A
A similar technique can be used to determine the expression for which is equal to bY therefore
When b= 1 then
When b= 2 then
Because can be expressed as then can be expressed like which can be expressed in a matrix form, like it is shown below.
=
However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, B. In order to find the final expression for , we must multiply the general scalar value , the final general expression being: B
To find the expression for a similar technique as before may be used; we know that A=aX and B=bX. From this we can develop an expression for:
=
The expression for is then
Example:
When a=-1
=
When b=2
=
When
Conclusion
M2=A2+B2
= (aX) 2+(bY)2
=+
=
To further illustrate that I have substituted integer values of a and b.
Let a= -1 and b= -1
Finally we are asked to come to a general statement for in terms of aX and bX. From all the calculation thus far we can deduce that therefore we can find the general statement by adding the expressions developed for :
=
To validate my general statement I must now substitute values for n, a and b:
a=4 and b= 6
A=
Using my general statement I should be able to come up with the same value for A+B:
= aX and = bX
=
, Proving that my general statement is indeed correct.
n=3 a = 10 and b = 3
Again using my general statement I should be able to get the same value for A+B:
= aX and = bX
, once againproving that my general statement is indeed correct.
Limitations/Scope
- As is one of the characteristics of matrices, fractions cannot be put as a power only integers can be powers. Thus using a non integer on the formula would not generate an appropriate answer.
- The general formula also doesn’t work for negative number as matrices can’t have negative powers.
- Another limitation is that A B and n must not equal each other.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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