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

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,

$B^2 = BB$

$⇒B^2 = (BA) B$  $[∵ BA =B]$

$⇒B^2=B (AB)$

$⇒B^2=BA$  $[∵ AB = A]$

$⇒B^2 = B$  $[∵ BA =B]$