Introduction

The goal of this portfolio assessment is to find an expression for Xn, Yn, (X + Y)n, [An, Bn, (A + B)n] in both questions whilst expressing them: Mn in terms of aX and bY. The purpose of this assessment is to find out how we can interpret matrix binomials using different values and similarities to find the pattern occurring. We’ve been given a general statement to express Mn in terms of aX and bY, to do so we must substitute a into matrix X to get a new matrix ‘A’, and b into matrix Y to get the new matrix ‘B’. The task given now is to see if the pattern really did work with other numbers, and to prove the general statement.

  • Question 1

Let X =  and Y =. Calculate X2, X3, X4; Y2, Y3, Y4.

By considering integer powers of X and Y, find expressions for Xn, Yn, (X + Y)n.

Alright now to calculate X2, X3, X4; Y2, Y3, Y4, I will firstly show how these matrices are multiplied, and then I shall use my graphics calculator to do the rest. As doing so I will also look for a pattern trend in which I can use to relate to fine the expression Xn, Yn, (X + Y)n. By doing so I will carefully look at how the matrix trend is created, therefore making it easier to find the expression.

I found that I can use a specific matrix property in order to find the expression as well as the arithmetic progression. This will ultimately determine how the expression is achieved and if it’s feasible. I will continue using the rule throughout the first question, however, I may need to change the expression for the next question based upon my findings.

The first thing I am going to do is to demonstrate how a matrix is multiplied; a step by step method and then I shall continue using the graphics calculator. In the end I will also add a few examples to show if the expression found is truly valid.

*just a brief note before I begin the project, every red colored matrix means that this number has been added as an extra to find the pattern more clearly. Every yellow matrix indicates the question given from the worksheet.

As you may notice, there are two differently colored matrices. This is because the last two matrices are extra help. I needed to add two more values to have a clearer view of the pattern occurring. As you can see I also showed the working out, however, I used a method of Xn x X, meaning that every time I add a number I just multiply the answer by the same matrix again, with the power increasing as I go along. Now I’m going to use the same method for Matrix Y.

Join now!

As you might have noticed, this matrix has the same digits as Matrix A, however, the second and third numbers are negative, this and is due to the  placement.

Now to find the expression of Xn, Yn, (X + Y)n, I found a clear pattern in the matrices developed.  Each matrix was multiplied by 2. However, the power of X and Y also play a role which influence what’s on the inside of the matrix, from that I started experimenting and I came up with solution. Since all the answers are multiples of 2 and gradually increase, one ...

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