$A^n = B^n$ does not imply $A = B$ for matrices

For square matrices \(A,B\) and \(n\ge2\), \(A^n=B^n\) does not imply \(A=B\). In \(\Mat_2(\RR)\), the matrix \(M=\matt 0 1 1 0\) satisfies \(M^2=I_2=I_2^2\), and \(M\ne I_2\).