# Matrix Binomials IA

Tenzin Zomkey                Maths SL Type 1

Maths Portfolio Standard Level

International Baccalaureate

Matrix Binomials

The main aim of this portfolio is to investigate the matrix binomials and observe and determine a general expression from the patterns that we obtain through the workings. Throughout the project, I shall be using solely matrices of 2 x 2 formations, and investigate the patterns I find.

1. To begin with, we consider the matrices X = and Y =.

The values of these matrices, each raised to the power of 2, 3 and 4 are calculated, as shown below;

X2 = X =                                Y2 =  x =

X3 = x =        and     Y3 = x  =

X4 = x =                   Y4 = x  =

It can be observed that all the matrices calculated above are in the form of 2 X 2, they are all square matrices. The corresponding diagonal elements are also observed to be the same. Since the matrices of each nth power can be seen to be the value of 1 less than the nth term, the general expression for the matrix Xn in terms of n is -

Xn =

And the general expression for Yn is –

Yn =

Likewise, the values of the matrix (X + Y), raised to the power 2, 3, and 4 is calculated to find its general expression.

The matrix: (X + Y) =   +  =

So, (X + Y) 2 = =

(X + Y) 3 =  x  =

(X + Y) 4 = x  =

And from the above, we can infer that the general expression for (X + Y) n is as follows,

(X + Y) n =

Proof:

Taking n as 3, the value is substituted ...