Transposition is Self-Inverse

Theorem
All transpositions are self-inverse.

Proof
Let $$\pi = \begin{bmatrix} k_1 & k_2 \end{bmatrix}$$ be a transposition.

Writing $$\pi \pi$$ in cycle notation gives $$\begin{bmatrix} k_1 & k_2 \end{bmatrix} \begin{bmatrix} k_1 & k_2 \end{bmatrix}$$

from which we see that $$k_1 \to k_2 \to k_1$$ and $$k_2 \to k_1 \to k_2$$.