eigenvectors + proving reflections

JamesGoh

Im aware that the plane of reflection for any 3x3 orthogonal matrix is the eigenvector for
the case λ= -1

Is the reason why we use λ= -1 to determine the plane of reflection is because the determinant of an orthogonal, reflection-transform matrix should be -1 by definition ?


Once the plane of reflection is determined, is it necessary to prove that for the case of λ= 1,
we get the positive/negative of the plane of reflection ?