Matrix Investigation

I will try to investigate in powers of matrices (2x2). Also, try to find a pattern, if there is one.

A=[pic]

Using my GCD calculator to raise matrix A to different powers

[pic]= [pic]

[pic]= [pic]

[pic]=[pic]

[pic]= [pic]

[pic]= [pic]

[pic]= [pic]

[pic]=[pic]

The pattern that I can see is that when the power of matrix A is an even number e.g. 2,4,6,8 then the result is [pic] the identity matrix. However, when the power is an odd number the matrix stays the same so [pic]

My prediction for [pic] matrix is: [pic]

Using the GCD calculator I checked my answer and it is correct. The determinant for this matrix A is -1 because (1x(-1)-0x3), that means that if we multiply A with the inverse of A so [pic] the result would be [pic] identity matrix.

[pic]= [pic] [pic] [pic] which basically shows us that the inverse of this matrix is the same as the original one.

A general rule for [pic](using algebra)

When the ‘n’ is an even number

[pic]= A[pic]

when the ‘n’ is an odd number

[pic]= A(A[pic]

It’s basically really simple one because of the determinant, which was -1, so when we make it as a fraction [pic] the result is still the same.

Now, I am considering the matrix B= [pic]

Using my GCD calculator I am calculating B raised to different powers.

[pic]= [pic]

[pic]= [pic]

[pic]= [pic]

[pic]= [pic]

[pic]= [pic]

The determinant of this matrix is -4 so probably the formula from before would not work because it’s not an identity matrix. But what we can see it is somehow related to the identity matrix. Because of the first result, which is just squaring, is 4x[pic]

From these calculations I can see that the formula for an even powers would be:

[pic]= [pic] so [pic]= [pic] = [pic]

[pic]= [pic] = [pic]

And when the power is an odd number det= -4

[pic]= [pic][pic] [pic] so [pic]= [pic]

= [pic]=[pic] [pic]= [pic] = [pic]=[pic] My prediction for [pic] would be

[pic]= [pic]

= [pic]=[pic]= [pic]

=[pic]

As I checked it using my GCD calculator and it is right we can consider that the formula is working for matrix B, which has a determinant equal to -4

Now I am trying to generalized this rule and try different values for a, b and n.

[pic]

Using the GCD

[pic]= [pic] [pic]= [pic]

Checking with the formula (the determinant is equal to -16)

[pic] So [pic] [pic]= [pic]

= [pic]= [pic]

Using the GCD and formula to see if the pattern is working:

[pic]=[pic] [pic] So [pic] (the determinant is equal to -9) [pic]=[pic] [pic]=[pic]=[pic]

[pic]=[pic]

[pic]= [pic]

The formula works so far, however now I am going to try raise matrix to a negative power and see, if the formula is working:

[pic] I can’t put it into the calculator. But we know that when we raise something to the negative power is the same as:

e.g. [pic] = [pic]

[pic]=[pic] [pic]

[pic]=[pic]

[pic]= [pic]

The rule for negative powers make sense, we would always end up with 1 over matrix. So simply saying when the n was a positive odd number the matrix was [pic] and when n was the same but negative the result was [pic] so almost the same but every element in the matrix was 1 over the result...