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 $k_2 \to k_1 \to k_2$.

The result follows from the definition of self-inverse.

$\blacksquare$