## Proof That Determinant of 2 x2 Matrices is a 2-Linear Function

Let
$A= \left| \begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array} \right|$
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.