Determinant of 2x2 Matrix

For a $2\times 2$ matrix, we have the formula

$$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=ad-bc$$

Proof

1. Let

a & b\\ c & d\\ \end{array}\right] ) = \det(a e_{1} + be_{2} + ce_{3} + de_{4})

2. $=a\det (e_1,c{e}_1,d{e}_2)+b\det (e_{21}{e}_{c_1}+d_{e_2})$
3. $=a[c\det (e_1,e_2)+d\det (e_1,e_2)]+b[c\det (e_2,e_1)+d\det (e_2,e_2)$
4. $=a[0+d]+b[-c+0]=ad-bc$