If A, B are two square matrices such that AB=A and BA=B, then

we have

$A B = A$ and $B A = B$

Now, $A B = A$

$\begin{aligned} \Rightarrow (A B) A = A A \\ \Rightarrow A (B A) = A^{2} \end{aligned}$

$\begin{array}{ll} \Rightarrow A B = A^{2} [∵ B A = B] \\ \Rightarrow A = A^{2} [∵ A B = A] \end{array}$

and, $B A = B$

$\begin{aligned} \Rightarrow (B A) B = B^{2} \\ \Rightarrow B (A B) = B^{2} \end{aligned}$

$\begin{array}{llrl} \Rightarrow B A = B^{2} [∵ A B = A] \\ \Rightarrow B = B^{2} [∵ B A = B] \end{array}$

$A^{2} = A$ and $B^{2} = B$

$\Rightarrow A$ and $B$ are indempotent matrices.

If $A, B$ are two square matrices such that $A B = A$ and $B A = B$ , then