# MATRIX BINOMIALS. In this investigation, we will identify a general statement by examining the patterns of the matrices.

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Introduction

Page

MATHEMATICS STANDARD LEVEL INTERNAL ASSESSMENT

MATRIX BINOMIALS

Summary of Investigation:

A matrix can be defined as a rectangular array of numbers of information or data that is arranged in rows and columns. There are a number of operations in which these matrices can perform (i.e., addition, multiplication, etc). In this investigation, we will identify a general statement by examining the patterns of the matrices.

Investigation:

Middle

= , = , =

Here, it seems reasonable to suggest a pattern for the X and Y values.

And so, by considering integer powers of X and Y, we can find the expressions for,:

= , =,

With the aforementioned expressions for the value of and, we will now determine the value for. This can be done through substituting the value of n to find a pattern for the matrices, as done so when determining the value of and.

Thus, with these patterns, the following expression can be suggested:

The matrices X and Y can now be used to form two new matrices A and B. Here, we will use a and b as constants for the matrices A and B, respectively. And hence the following:

,

Now, the different values of a and b can be used to calculate the values of

And therefore,

, ,

, ,

With the patterns from these matrices, we can determine the expressions for matrices A and B by considering its integer powers:

We will now investigate a new matrix,

Conclusion

The Algebraic Method:

Lastly, we will investigate the use of an algebraic method to explain how the general statement was reached.

To begin with, we let A=aX and B=bY, where X= and Y=

A=a= , B=b=

Now, we let , and A= and B=

A+B= ,

and

And therefore,

M=A+B

And, if Mn = (A+B)n,

,

And given that,

and

It seems reasonable to suggest the general statements,

(A+B)n = An + Bn

Mn = An + Bn = (aX)n+(bY)n=anXn + bnYn

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

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