Find a 2 by 2 matrix such that when cubed, is equal to the identity matrix

[theBEAST]

Homework Statement

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]

The Attempt at a Solution

The professor told us that we have to use a linear transformation where you rotate it three times by 120^o. 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!

 

[vela]

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

 

[theBEAST]

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

 

[vela]

Think "rotation matrix."

 

[Deveno]

Homework Statement

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]

The Attempt at a Solution

The professor told us that we have to use a linear transformation where you rotate it three times by 120^o. 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!

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.