# Homework Help: Matrix to an exponent

1. Oct 9, 2012

### DmytriE

1. The problem statement, all variables and given/known data
Consider the matrix

cos(3*pi/17) -sin(3*pi/17)
S = sin(3*pi/17) cos(3*pi/17)

Does there exist a positive integer n such that Sn = I where I is the 2x2 identity? If so, what is the smallest such integer? Explain.

Excuse the poor matrix formatting. I do not know how to use the latex formatting to put it into pretty print.

2. Relevant equations
Not sure....

3. The attempt at a solution

Where should I start? I really have no idea.

2. Oct 9, 2012

### tiny-tim

Hi DmytriE!

Hint: suppose S is

Code (Text):
cosθ -sinθ
sinθ cosθ
What is S2 ? S3 ? etc?

3. Oct 9, 2012

### DmytriE

Let's suppose that n = 10. I don't think I have to multiply S by S 10 times to get the answer. Unfortunately the answer is not 10. How would S2, S3 help me figure it out?

S2:
UL:cos2(θ) - sin2(θ)
UR: -2sin(θ)cos(θ)
LL: 2sin(θ)cos(θ)
LR: -sin2(θ)+cos2(θ)

Each abbreviation represent the place in the matrix that they would appear. UL - Upper left, etc.

S3:
Alot of sines and cosines.

4. Oct 9, 2012

### Staff: Mentor

That's not what tiny-tim is suggesting. Your matrix represents a certain kind of transformation.

Instead of thinking about what S, S2, S3, etc. are (in terms of their matrix representations), think about what they do to a vector they multiply.

5. Oct 9, 2012

### DmytriE

Thanks for the help! This forum really is the best!

6. Oct 10, 2012

### tiny-tim

Hi DmytriE!

(just got up :zzz:)
have you got it now?

if not, use standard trigonometric identities