Math IA - Matrix Binomials

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IBO INTERNAL ASSESSMENT

Matrix Binomials

Mathematics SL Type I

Katie Xie

Mrs. Cheng

9/23/2008

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

X2=X ? X

X2=

X2=

X2=

X3=X2 ? X

(Matrix multiplication is associative)

X3=

X3=

X3=

X4=X3 ? X

X4=

X4=

X4=

Y2=Y ? Y

Y2=

Y2=

Y2=

Y3=Y2 ? Y

(Matrix multiplication is associative)

Y3=

Y3=

Y3=

Y4=Y3 ? Y

Y4=

Y4=

Y4=

As we can see, a general trend emerges as we increase the power of the matrix. There is a definite relationship between the power of the matrix and the end product (entries in the matrix). We observe that when X is to the power of 2, i.e. X2, the matrix's entries are all 2's; when X3, the entries are all 4's; when X 4, the entries are all 8's.

For the Y matrix, a similar pattern emerges, except in this case, we must note the negative signs. However, these two negative numbers always occupy the