I need to pull out, for example, row 2 of a matrix, but never add the elements in a row, so I believe a Kronecker product involving an augmented matrix containing identity matrices is required. The operation would be something like the following:
$ \pmatrix{ x_{12} & x_{12}\cr x_{21} & x_{22}\cr x_{31} & x_{32}\cr x_{41} & x_{42}\cr } \otimes \pmatrix{ \bf{A} & \bf{B} & \bf{C} & \bf{D}\cr \bf{E} & \bf{F} & \bf{G} & \bf{H}\cr } = \{x_{21}, x_{22}\}$
The resulting matrix needs to contain only a single copy of $x_{21}$ and a single copy of $x_{22}$ with all other elements equal to zero. (NB: I am using the set of elements in row 2, $\{x_{21}, x_{22}\}$, to denote one copy of each in the final matrix).
I hope that I didn't get you wrong! You should do it by multiplying from left:
$\left( \begin{matrix} 0 & 1 & 0 & 0 \\ \end{matrix} \right)\left( \begin{matrix} {{x}_{11}} & {{x}_{12}} \\ {{x}_{21}} & {{x}_{22}} \\ {{x}_{31}} & {{x}_{32}} \\ {{x}_{41}} & {{x}_{42}} \\ \end{matrix} \right)=\left( \begin{matrix} {{x}_{21}} & {{x}_{22}} \\ \end{matrix} \right)$