Well it's not that the square is a negative number but rather that you were only taught to do arithmetic with 1x1 matrices so you don't have a nice abstraction

Well it's not that the square is a negative number but rather that you were only taught to do arithmetic with 1x1 matrices so you don't have a nice abstraction in your head to convince yourself. It's similar to people who insist that 0 is not a natural number because they haven't studied abstract algebra and don't care about the underlying algebraic structure (nevermind that peano arithmetic starts by postulating zero and succ...)