# Math's Portfolio SL Type 1 "Matrix Powers"

Newton College

Math’s Portfolio

SL Type 1

“Matrix Powers”

Mauro Gelmi

IB 2

March 2009

Introduction

Matrices are tables of numbers or any algebraic quantities that can be added or multiplied in a specific arrangement. A matrix is a block of numbers that consist of columns and rows used to present raw data, store information or to perform certain mathematical operations

The aim of this portfolio is to find a general trend in different sets of matrices, therefore find a general formula that applies for all of the matrices. The pattern will then be explained and tested to see if applies correctly to all the sets of matrices.

Method

To calculate Mn for n = 2, 3, 4, 5, 10, 20, 50 I used a GDC.

Determinants are defined as ad-bc in a 2  2 matrix,  and is denoted by det A = | A |. In Other words, det A = | A | =  = ad-bc.

M2 = =, det (M2) = 16 = 42

M3 = =, det (M2) = 64 = 43

M4 =   =, det (M2) = 256 = 44

M5 =    =, det (M2) = 1024 = 45

M10 =, det (M2) = 1048576 = 410

M20 =, det (M2) = 1.099511628 x 1012 = 420

M50 =, det (M2) = 1.2676506 x 1030 = 450

By squaring each number in the matrix by the power of Mn you get the answer for each of the matrices. So if M2 =, you then square the matrix, so you multiply it by itself. .        Multiplying the matrix by itself as this: gives you the new matrix which is, in this case.

Here, in the case of matrix Pn, if we divide each number in the resultant matrix after replacing n, by 2n-1 we notice a pattern in the numbers inside these new matrices. To calculate this we use a GDC.

P2 === 2, det (P2) = 64 = 82 (here k is 8-4)

P3 = == 22, det (P3) = 512 = 83 (here k is 8)

P4 = == 23, det (P4) = 4096 = 84 (here k is 8×2)

P5 = == 24, det (P5) = 32768 = 85 (here k is 8×4)

Given these matrices, we can notice that when the matrix is simplified by 2n-1 you get a new matrix which the numbers can be found by using the number multiplying it as k in the general formula. A pattern between the matrices’ determinants is noticeable. The determinant of each matrix is 8 powered by the same number you powered the matrix. ...