Notation for removal of row / column from matrix Is there some common notation for the result of removing the $i$th row, the $j$th column or both of them from a matrix given $A$?

How short do you need it? What's also often done is to use selection matrices that select a subset of rows/columns. E.g., to drop the third out of five rows of $A$ you can use $J_{3,5} \cdot A$ where $$J_{3,5} = 
\begin{bmatrix} 
  1 & 0 & 0 & 0 & 0 \\ 
  0 & 1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 1
\end{bmatrix}.$$
Once you defined what $J_{n,p}$ means in general you can also drop rows and columns via $J_{n,p} \cdot A \cdot J_{m,q}^{\rm T}$ (where $A$ is $p\times q$), apply them sequentially àla $J_{k,p-1} \cdot J_{n,p} \cdot A$ etc.
