The matrix inverse “undoes whatever the matrix did” is pretty much the most straightforward interpretation, if you’re thinking of a matrix as performing some ki

The matrix inverse “undoes whatever the matrix did” is pretty much the most straightforward interpretation, if you’re thinking of a matrix as performing some kind of rotation/shear/scaling of space. There are some other interpretations, for example if you think of a linear system Ax=b as a kind of “checking machine” (does x satisfy Ax=b? Yes/no, try again), then the inverse of A can produce solutions, since x=(A inverse)b.