# Math IA - Matrix Binomials

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 same position in the matrix when the power is increased.

Through consideration of the integer powers of X and Y, we can now find expressions for Xn, Yn, (X+Y)n.

We observe that the elements of the matrices appear to form a geometric sequence; thus we can use the general equation for a geometric sequence to determine the expressions for Xn, Yn, (X+Y)n:

, where Un=a specific term

a=first term

r=common ratio (multiplier between the entries of the geometric sequence)

n=the number of the specific term (with relation to the rest of the sequence).

For Xn:

When n=1, 2, 3, 4, … (integer powers increase), then the corresponding elements of each matrix are:

1, 2, 4, 8, … These terms represent the pattern between the scalar values multiplied to

X= to achieve an end product of Xn.

Thus, we can now deduce the geometric sequence of these scalar values using the general equation listed above:

In the sequence {1, 2, 4, 8}, a=1

r=2

However, we must note that this formula only gives us the progression for the scalar values which will be multiplied to the matrix, X. In order to find the final expression for Xn, we must multiply the general scalar value 2n-1 by matrix X:

In order to test the validity of this expression, we can employ it to find X5. Using the previous method of calculations, we find that X5=. (We can also check this on a calculator).

Now, using the expression, we find the value of X5 to be the same (proving the accuracy of our expression):

X5=25-1

=24

=16

=

We conclude that is indeed a valid expression.

We can find an expression for Yn in a similar fashion.

, where Un=a specific term

a=first term

r=common ratio (multiplier between the entries of the geometric sequence)

n=the number of the specific term (with relation to the rest of the sequence).

For Yn:

When n=1, 2, 3, 4, … (integer powers increase), then the ...