## Nulti Linear Function Example

A commutative ring with an identity satisfies all the axioms for a ring field except the requirement that every element has a multiplicative inverse. We use this structure when there is no need for division.
Define n - linear functions from the domain of n x n linear matrices to the field
$K$
. Let
$D$
be a function which assigns to every n x n matrix
$A$
over
$K$
a scalar
$D(A) \in K$
.
$D$
is n - linear if
$1 \lt i \lt n$
,
$D$
is a linear function of the ith row.
Now let
$D$
be a 2 - linear function. Let
$e_1, \; e_2$
be the rows of the 2 x 2 identity matrix, then
\begin{aligned} D(A) &= D(A_{11} e_1 + A_{12}e_2,A_{21}e_1+A_{22}e_2)\\ &= A_{11} D(e_1,A_{21}e_1+A_{22}e_2) +A_{12} D(e_2,A_{21}e_1+A_{22}e_2) \\ &=A_{11}A_{21} D(e_1,e_1)+A_{11}A_{22} D(e_1,e_2)+A_{12}A_{21} D(e_2,e_1)+A_{12}A_{22} D(e_2,e_2) \end{aligned}

This
$D$
is determined by its effects on the rows of the identity matrix.
Let
$a, \; b, \; c, \; d \in K$
be any four scalars. Define
$D((A)=A_{11}A_{21}a+A_{11}A_{22}b+A_{12}A_{21}c+A_{12}A_{22}d$

Then
$D$
is a 2 - linear function on 2 x 2 matrices and
$D(e_1,e_1)=a, \; D(e_1,e_2)=b, \; D(e_2,e_1)=c, \; D(e_2,e_2)=d$