Transformation of coordinates axes

Question

Show that there is one point whose co-ordinates do not alter due to a rigid motion.

Answer 1

We need to show that exist x fixed point such that

x = Rx + x₀

⇒ (I−R)x = x₀

and thus that

for x₀ = 0

⇒ x = 0 (no translation)

or

for x₀ ≠ 0 

(I−R) is invertible and x = (I−R)⁻¹ x₀