So it's been a while since I've taken Linear Algebra, but my friend asked me a question, that I couldn't answer.
If a matrix $A$ exists such that $A^3 = I$, does $A$ have to equal the identity matrix $I$?
My first instinct was to say no, but... (edited out my incorrect math)
EDIT: thanks guys for the awesome examples
EDIT2: Followup question: Is there a way to solve for all possibilities of A if given A^3 = I?