Let A be an orthogonal 3 × 3 matrix with real entries. Pick out the true statements:
1
The determinant of A is a rational number.
2
d(Ax, Ay) = d(x, y) for any two vectors x and y in ℝ3, where d(u, v) denotes the usual Euclidean distance between vectors u, v ∈ ℝ3.
3
All the entries of A are positive.
4
All the eigenvalues of A are real.