# Matrix Binomials Portfolio

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Introduction

Math SL Matrix Binomials Portfolio

This portfolio will investigate the properties of matrix binomials in order to determine a general statement for Mn where n is a real number and an integer, and M is the 2 matrix

Let:

X = Y=

X2 = = X3 = =

X4 =

Y2 = Y3 =

Y4 =

(X+Y) =

(X+Y)2 = (X+Y)3 =

(X+Y)4 =

Expressions for Xn, Yn and (X+Y)n

Xn = Yn = (X+Y)n =

n > 0,

let: W = any 2x2 matrix,

W-n = → It is not possible to

Middle

B3 =

B4 =

Therefore:

(A+B) =

(A+B)2 =

(A+B)3 =

(A+B)4 =

Expressions for An, Bn and (A+B)n

An = Bn =

(A+B)n =

n > 0,

let: W = any 22 matrix,

W-n = → It is not possible to divide an integer by a matrix, n < 0 does not exist

n≠0

For any matrix where n=0 Wn = I W0 =

Let: M = , M = A+B and M2 = A2+B2

A = aX =

Conclusion

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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