I have a question about generator and parity matrices. Let's say I have a code $C$ with generator matrix $G$. I want to find $H$, its parity check matrix. I want the parity check matrix of $C$, not of a code equivalent to $C$. To get $G$ in standard form I can perform several operations. I know that those that affect the rows of the matrix won't change the code that the matrix generates, but if I multiply a column by a non-zero scalar, or if I do a permutation of columns, the code my matrix generates will change (although it will be equivalent to $C$, but this is not what I want).
Let's say I perform the required operations and I get my matrix $H$. How can I obtain a parity check matrix for $C$? I know that I must undo the permutations, but what about the multiplication of columns by scalars? How do I reverse that operation?