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\[\begin{equation} \begin{aligned} D(A) &= D(A_{11} e_1 + A_{12}e_2,A_{21}e_1+A_{22}e_2)\\ &= A_{11} D(e_1,A_{21}e_1+A_{22}e_2) +A_{12} D(e_2,A_{21}e_1+A_{22}e_2) \\ &=A_{11}A_{21} D(e_1,e_1)+A_{11}A_{22} D(e_1,e_2)+A_{12}A_{21} D(e_2,e_1)+A_{12}A_{22} D(e_2,e_2) \end{aligned} \end{equation}\]
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\[D(e_1,e_1)=a, \; D(e_1,e_2)=b, \; D(e_2,e_1)=c, \; D(e_2,e_2)=d\]