Let $A \in \mathbb{C}^{2 \times 2}$ be a
D6159: Complex square matrix such that
| (i) |
\begin{equation}
A
=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\end{equation}
|
The standard permutations in $S_2$ are $(1, 2)$ and $(2, 1)$. Result
R5521: Signs of length-2 standard permutations shows that $\text{Sign}(1, 2) = 1$ and $\text{Sign}(2, 1) = -1$. Therefore, we have
\begin{equation}
\begin{split}
\text{Det} A
& = \sum_{\pi \in S_2} \left( \text{Sign}(\pi) \prod_{n = 1}^2 A_{n, \pi(n)} \right) \\
& = \text{Sign}(1, 2) A_{1, 1} A_{2, 2} + \text{Sign}(2, 1) A_{1, 2} A_{2, 1} \\
& = A_{1, 1} A_{2, 2} - A_{1, 2} A_{2, 1} \\
& = a d - b c
\end{split}
\end{equation}
$\square$
