Or just memorize it: 11:00
At 9:14, the reason it is only defined when A is square is because you can only take inverses of square matrices! I misspoke when I said the inverse of a 4x3 would also be 4x3; the inverse of a 4x3 matrix is not defined.
Of course, you can always construct the standard matrix by seeing where the columns of the identity matrix end up, but that gets really hairy trying to project onto 3 or 4 dimensional subspaces. Then again, that could still be less hairy than multiplying all those matrices together, so it's just a pick your poison kind of thing I guess lol
