# Determinant 2x2

Show the determinant of a 2x2 matrix
is

### Proof

By definition of determinant, the determinant of a 2x2 matrix
is
where $\sigma$ is a one-to-one mapping from a set $\{1,2 \}$ to the same set. There are two cases that
and
Let $\sigma_{12}$ be the above case and $\sigma_{21}$ be the below case. $S_{2}$ is a set having $\sigma_{12}$ and $\sigma_{21}$ as elements,
$\mathrm{sgn}(\sigma)$ is a sign assigned to $\sigma_{12}$ and $\sigma_{21}$ as
($\mathrm{sgn}(\sigma)$ becomes negative if the order of number $\{1,2\}$ is changed odd times.)
Therefore $|A|$ is written as
Example
Using the above result, the determinant of a 2x2 matrix
is
Calculator
Enter a 2x2 matrix and press "Execute" button. Then its determinant is displayed.
 1 2 1 2
$|A|$ =