Given that:

and where a and b are constants

We were asked to find using different values of a and b and then find the expressions for .

Therefore:

And:

Assuming that a = 2

Assuming that a = 3

Assuming that a = 6

Assuming that a = -4

Assuming that b = -1

Assuming that b = 4

Assuming that b = 5

Assuming that b = 7

To find the expression for I observed that the final results achieved were always the value chosen for a multiplied by the matrix X and the product to the power of n.

For example (assuming a = 2):

As a result we can see that:

And since we know

Hence:

To find the expression for I did the same as for changing the value a to b and X to Y.

For example (assuming b = -1):

As a result we can see that:

And since we know

Hence:

To find I used the binomial theorem

From the binomial theorem we can see that the values of A and B multiply by each other on every term except the first and the last, where we find.

However if we multiply matrix A by B we will see that the product will be a zero matrix.

This allows us to cancel every term in which A multiplies B or vice-versa, as the result will be zero.

Therefore if we cancel these terms we will be only left with.

This means that

Now we were given:

And asked to prove that

So:

A B M

To find, first we need to calculate A², B² and M².

So:

Therefore:

To find the general statement that expresses in terms of aX and bY we first need to continue the sequence and find and .

So:

By analyzing all 4 values for we can see that the result can be put into multiplied by 2a+2b in a sequence.

Example:

Hence we can say that our general statement is:

Testing the validity of my general statement, to do this we had to get the same results for both expressions:

Assuming: a = 2, b = 4, n = 1

Assuming: a = -1, b = -3, n = 1

Assuming: a = -5, b = 2, n = 1

Assuming: a = 1, b = 3, n = 2

Assuming: a = 1/2, b = 2, n = 1

Assuming: a = 2, b = 2, n = -2

Assuming: a = 2, b = 2, n = 0

Not Compatible

After testing the validity of my general statement we can see that the results for both formulas were mostly compatible for all numbers of a, b and positive n, proving the validity of our statement. However when n is zero or a negative integer we find some problems with it. As we can’t power a matrix to a negative number n can’t be a negative number and when 0 we find out all matrices to the power of 0 form identity matrices that when added in the formula it differs from our other result which is a identity matrix as well.

To get to this formula algebraically I did:

This can be concluded by:

By knowing that AB =0:

And finally I found the general expression: