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

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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:

To begin with, we are given the two matrices, X and Y.

We let X=and Y=, and calculated

= ==

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====

===

====

===

=

And therefore,

= , = , =

= , = , =

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 ...