Is there a formal way of algebraically expressing ( for example as a product of specially designed matrices) - duplicating specific row/column(s) or alternatively omitting rows/columns ?
for example I want to able to relate matrices A and B, where $$ A= \left[ \begin{array}{ccc} a&a&a&b&b\\ a&b&b&a&b \end{array} \right] $$ and $$ B=\left[ \begin{array}{ccc} a&b&b\\ b&a&b \end{array} \right] $$ where the only operation is omission or duplication of vectors.
Could be a simple operation but it's beyond me.
Here's a rule of thumb: if you want a matrix that does something to the rows via left-multiplication, do that thing to the identity matrix. If you want a matrix that does something to the columns via right-multiplication, do that thing to the identity matrix.
For example: say we want to duplicate the second row of a matrix with $3$ rows. The associated matrix would be $$ R= \pmatrix{1&0&0\\ 0&1&0\\ 0&1&0\\ 0&0&1} $$ which is simply the size-3 identity matrix with a duplicated second row.