How does this question on matrix multiplication work?

Question

So, there was this solved question in my reference book- $$ AB = A \space \space \space \space BA=B$$ And I was given the following options- $$1. A = I$$ $$2. B = I$$ $$3. A = A^2$$ $$4. B = B^2$$

And according to the book only the last two are true. But shouldn't the first two be true as well? since, $$AB = A$$ $$A^{-1}AB = I$$ $$B = I$$ Similar should be true for $A$ by symmetry.

Answer 1

Hint:

Would the first two options be correct if $A, B$ were singular?