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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:

...read more.

Middle

image37.png=image39.png , image41.png=image43.png , image44.png=image46.png

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 forimage48.png,image50.png:

image48.png= image51.png, image50.png=image52.png,

With the aforementioned expressions for the value of image48.pngandimage50.png, we will now determine the value forimage53.png. 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 image48.pngandimage50.png.

image54.png

image55.png

image56.png

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

image57.png

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:

image58.png, image59.png

Now, the different values of a and b can be used to calculate the values of image60.png

image61.png

image62.png

image63.png

image04.png

image64.png

image65.png

And therefore,

image66.png, image67.png, image68.png

image69.png, image70.png, image71.png

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

image72.png

image73.png

We will now investigate a new matrix, image26.png

...read more.

Conclusion

n were not included because it lacked its purpose when in a matrices. Furthermore, although a matrix may be raised to the power of -1, it does not identify an exponent; but rather, the inverse of the matrix. In this case, if the matrix is multiplied with another matrix, the value will still be equivalent to the original matrix. Therefore, it seems reasonable to suggest that the general statement can be applied when the determinant is not equal to zero. However, because there may be possible abnormalities, such as the identity matrix, there seemed to be a limit when investigating the general statement.

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=image00.png and Y= image01.png

A=aimage00.png= image20.png, B=bimage01.png= image21.png

Now, we let image22.png, and A=image23.png and B= image24.png

 A+B= image25.png,

image26.pngand

And therefore,

 M=A+B

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

image27.png,image28.png

And given that,

image29.pngand image30.png

It seems reasonable to suggest the general statements,

         (A+B)n = An + Bn

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

...read more.

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