Question

My problem looks like this: `((2, -1), (3, -2))^n` it's a 2x2 matrix. Can someone correct me?

My answer:
for all n- even the matrix will look like this `((1, 0), (0, 1))`
for all n- odd the matrix will look like this `((2, 3), (-1, -2))`
Do I need to prove this, if so how?

Answer 1

See explanation...

Explanation:

Note that:

`((2, -1), (3, -2))^2 = ((2, -1), (3, -2))((2, -1), (3, -2)) = ((1, 0), (0, 1))`

So:

`((2, -1), (3, -2))^(2k) = ((1, 0), (0, 1))^k = ((1, 0), (0, 1))`

and:

`((2, -1), (3, -2))^(2k+1) = ((1, 0), (0, 1))^k((2, -1), (3, -2)) = ((1, 0), (0, 1))((2, -1), (3, -2)) = ((2, -1), (3, -2))`