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Clash Royale CLAN TAG #URR8PPP 3 1 How do I prove that, if for a $2 times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= beginbmatrix cosx& -sinx\ sinx & cosxendbmatrix ,$$ for some real $x$ , then $A$ is also a rotation matrix? I know that it is very obvious, especially if you think about the isomorphism with complex numbers, but I can't seem to come up with a simple and rigorous proof. linear-algebra rotations share | cite | improve this question edited Dec 16 at 22:51 amWhy 191k 28 224 439 asked Dec 16 at 15:08 Tanny Sieben 301 1 8 add a comment  |  3 1 How do I prove that, if for a $2 times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= beginbmatrix cosx& -sinx\ sinx & cosxendbmatrix ,$$ for some real $x$ , then $A$ is also a rotation matrix? I know that it is very obvious, especially if you think about the isomorphism with complex numbers, but