Now an inverse intuitively means as shown in the diagram below:
How I've understood is that it is a matrix that it transforms a vector transformed by A back to it's original position, if A is the standard transformation matrix then,
There are two questions that come to me:
1) Why should A be row equivalent to the Identity matrix I? (Even more confusing is that it's product with it's inverse should be I), where's the intuition?
2) Is being row equivalent a sufficient condition or a necessary condition or the one and only condition? (I don't see this stated much anywhere)