If $AB = A$ and $BA = B$, where A and B are square matrices, then

$B^2 = B$ and $A^2 = A$ $B^2 ≠ B$ and $A^2 = A$ $A^2 ≠ A, B^2 = B$ $A^2 ≠ A, B^2 ≠ B$

$B^2 = B$ and $A^2 = A$

We have, $A^2 = AA$ $⇒A^2 (AB) A$  $[∵ AB=A]$ $⇒A^2 = A (BA)$ $⇒A^2 = AB$  $[∵ BA=B]$ $⇒A^2 = A$   $[∵ AB = A]$ and, $B^2 = BB$ $⇒B^2 = (BA) B$  $[∵ BA =B]$ $⇒B^2=B (AB)$ $⇒B^2=BA$  $[∵ AB = A]$ $⇒B^2 = B$  $[∵ BA =B]$