# Homework Help: Find a 2 by 2 matrix such that when cubed, is equal to the identity matrix

1. Feb 22, 2012

### theBEAST

1. The problem statement, all variables and given/known data
Find a 2 by 2 matrix such that when cubed, is equal to the identity matrix. This matrix cannot be equal to the identity matrix unless it is cubed.

So for example:
B3 = [1 0;0 1]
but
B≠[1 0;0 1]

3. The attempt at a solution
The professor told us that we have to use a linear transformation where you rotate it three times by 120o. The problem I have is that I cannot visualize how such rotations can solve the problem. Also I don't even know what to rotate. If anyone knows what to do it would be greatly appreciated, thanks!

2. Feb 22, 2012

### vela

Staff Emeritus
What happens when you rotate a vector x by 120 degrees about the origin three times?

3. Feb 22, 2012

### theBEAST

You get back to the original vector. But i still can't relate this to cubing a matrix.

4. Feb 22, 2012

### vela

Staff Emeritus
Think "rotation matrix."

5. Feb 22, 2012

### Deveno

what you would be rotating is the entire plane, or R2. such a rotation would take the point (x,y) = (1,0) to the point (x',y') = (cosθ,sinθ). that "almost" gives you the matrix right there. use geometry to see if you can figure out where (0,1) might rotate to.