\[A= \left| \begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array} \right|\]
and define \[D(A)=A_{11}A_{22}-A_{12}A_{21}\]
/\[D\]
is the sum of the 2 - linear functions \[D_1=A_{11}A_{22}, \; D_2=-A_{12}A_{21}\]
, so is itself a linear 2 - function, so the determinant of 2 x 2 matrices is a 2 - linear function.