Prove, by construction, that the elements of SO(2) and SO(3) do not have a unique inverse. I know the properties of the Matrix Group SO(n) are

• R ∈ SO(n)

• R^−1 ∈ SO(n)

• R^−1 = R^T

• The columns (and therefore the rows) of R are mutually orthogonal 

• Each column (and therefore each row) of R is a unit vector

• det R = 1

How can I prove by construction, that the elements of SO(2) and SO(3) do not have a unique inverse?
Imagine a group $G$, and suppose there exist two different elements $h_1,h_2\in G$ that are inverse of the element $g\in G$, that is $gh_1 = e$, and $gh_2 = e$, with $e$ the identity element of $G$. You have then

$$
h_1 = h_1 e = h_1 (g h_2) = (h_1 g)h_2 = e h_2 = h_2
$$

that is, $h_1 = h_2$ which contradicts the assumption $h_1 \not= h_2$. We have just proved that the inverse is unique. As such, the inverse of any element in $SO(2)$ or $SO(3)$ is unique!
The groups SO(2) and SO(3) are the group of rotational matrices in 2-D and 3-D, respectively.
By definition a 3 by 3 matrix, $R$, belongs to SO(3) group if $R^T.R=I$ and $det(R)=1$. In other words, $R$ is a 3 by 3 orthogonal matrix with $det(R)=1$. 

If $R∈SO(3)$ then $R^(-1)∈SO(3)$. Having said that, Equation 1 shows that if an object is rotated by an arbitrary angle $σ$, then the original object can be recovered either by rotating by $–σ$ or rotating by $2π-σ$. 

$$(2π)k-σ, \quad  k∈{0,1,2,…,n}   \qquad (Equation 1)$$


If the rotational matrices of $$–σ$$ and the scalar multiples of $$2π-σ$$ are treated to be the same, the uniqueness of inverse matrices can be proved as follows:


Let $A$ has two inverses, denoted by $B$ and $C$, in the group SO(3) (or SO(2)).
By definition of inverse matrices,

$$AB=I \quad \textrm{and} \quad BA=I$$

Likewise,

$$AC=I \quad \textrm{and} \quad CA=I$$

Then,

$$B=BI$$

$$B=B(AC)$$

$$B=(BA)C$$

$$B=IC$$

$$B=C     \qquad  (Equation 2)$$

Equation 2 shows that $B$ and $C$ are equal hence there is only one inverse element of $A$, i.e. the matrix $A$ has a unique inverse.


If the rotational matrices of $–σ$ and the scalar multiples of $2π-σ$ are not treated to be the same then this proves that the matrix $A$ has multiple of inverses in the group SO(3).


Since the question states that "the elements of SO(2) and SO(3) DO NOT have a unique inverse." the second bullet point was chosen.