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  
. Let  
  be a function which assigns to every n x n matrix  
  a scalar  
\[D(A) \in K\]
  is n - linear if  
\[1 \lt i \lt n\]
  is a linear function of the ith row.
Now let  
  be a 2 - linear function. Let  
\[e_1, \; e_2\]
  be the rows of the 2 x 2 identity matrix, then
\[\begin{equation} \begin{aligned} D(A) &= D(A_{11} e_1 + A_{12}e_2,A_{21}e_1+A_{22}e_2)\\ &= A_{11} D(e_1,A_{21}e_1+A_{22}e_2) +A_{12} D(e_2,A_{21}e_1+A_{22}e_2) \\ &=A_{11}A_{21} D(e_1,e_1)+A_{11}A_{22} D(e_1,e_2)+A_{12}A_{21} D(e_2,e_1)+A_{12}A_{22} D(e_2,e_2) \end{aligned} \end{equation}\]

  is determined by its effects on the rows of the identity matrix.
\[a, \; b, \; c, \; d \in K\]
  be any four scalars. Define

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

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