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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.

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    Can you edit in an example of what you are trying to do? Thanks in advance! – 2017-01-22
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    I always use something along the lines of $R_m := R_n$ – 2017-01-22

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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.