linear algebra - Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B

I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations are relatively simple. I'd like to understand why/ how this solves the system of linear equations.

I know my question is asking more (or arguably less) than a concrete sequence of steps, a theorem, etc. But I think someone who understands linear algebra much better than I can get through to me better than my texts' treatment, which is little more than a worked example.

Thanks in advance..

Answer

By doing the row operations that change $[A|I]$ to $[I|B]$ stores up the row operations necessary to "untangle" $A$. This compiles those operation into the matrix $B$. You can think of $B$ as an executable program that undoes $A$.