Mathematical Thinking - Matrices

[yarnlife]

Hi guys, iv been stuck on this problem for a while now and can't seem to make any headway

construct a counterexample to the following statement:

"for matrices A with real entries 'A^3=Identity impies A=Identity'

im not restricted by size for the matrix.

any hints would help because i just can't think of any counterexamples!

thankyou

 

[Evgeny.Makarov]

You may consider $\begin{pmatrix}\cos\varphi&-\sin\varphi\\\sin\varphi&\cos\varphi\end{pmatrix}$, which is the matrix of rotation on the plane by angle $\varphi$ around the origin.