Math IB portfolio assignment - MATRICES

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Ankit Shahi

Investigating Matrix Binomials

Introduction to Matrix Binomials

Matrix Binomials can be defined as a type of a 2 by 2 matrix. Generally speaking, matrix binomials come in the form  . These matrix binomials can be defined as the sum of two component matrices. One component should be known as the positive matrix. All elements within the positive matrix have the same positive value. The other part should be called as the negative matrix. All elements within the negative matrix have the same magnitude but the top right and bottom left elements have a negative value.

The overall goal of this project is to investigate the properties of these matrix binomials in relation to its positive and negative matrix components.

The first step would be investigating the positive and negative matrix components separately as they are the simplest components. We shall begin by defining X and Y as the simplest positive and negative matrices respectively and finding their general expressions.

Let  

From these experimentations with the matrix X, we notice a clear pattern. In each consecutive matrix, the values of all four elements are twice as great as the values within the previous matrix. Therefore, as the power of X is increased by one, the values of all the elements within the matrix are multiplied by two. This trend is understandable since the process of matrix multiplication is row by column. Since there are two rows and columns in each column, the sum of the products of the first elements and the second elements is two times the original value.

Let

In each consecutive matrix, the values of all four elements are twice as great as the values within the previous matrix. Therefore, as the power of X is increased by one, the values of all the elements within the matrix are multiplied by 2. It should be noted that all of the elements remained positive.

Table 1: Matrices for Xn and Yn

Now that we have an idea of different patterns when X and Y are raised to an exponent ranging from 1 to 4, we can create a table for different values of Xn and Yn with higher exponents. Using the graphing calculator, X and Y were entered through the Edit Matrix window. The expressions for Xn and Yn were found by entering [X] or [Y}, ^ and n on the home screen. The different matrices are shown to the left in Table #1.

The general expression for Xn seems to be. This can be derived from the fact that X doubles every time n increases. Since X1, the starting power of 2 needs to be one less.

 For Yn, it seems to be. The same rules apply to Y as they do to X. The key difference is that two elements within the Y matrices maintain their negative signs.

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Having determined the basic general equations for both X and Y, we can combine X and Y into our simplest matrix binomial and attempt to find some interesting properties.

Once again, the pattern seems to be that each element in every consecutive matrix doubled. Of course, when the initial element is zero, all the elements of the same row and column equal zero.

However, the most important property that we notice is that (X+Y)n = Xn + Yn. We can see that this property holds true for all four trials done so far. The next step would be to test more ...

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