If A = [x y z; y z x; z x y] where x, y, z are integers, is an orthogonal matrix, then what is A² equal to?

For an orthogonal matrix, A·Aᵀ = I. Notice that A given here is symmetric (because the matrix [x y z; y z x; z x y] — checking: A[1][2]=y, A[2][1]=y ✓; A[1][3]=z, A[3][1]=z ✓; A[2][3]=x, A[3][2]=x ✓). So A = Aᵀ. Hence A² = A·A = A·Aᵀ = I, the identity matrix.