Question

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

• A. A, B are idempotent
• B. only A is idempotent
• C. only B is idempotent
• D. none of these

Answer 1

Answer: (A)

We have AB = A and BA = B
Now AB = A$\Rightarrow$ (AB) A = A.A
$\Rightarrow$A (BA) = A${}^2$ $\Rightarrow$ AB = A${}^2$ [$\because$BA = B]
$\Rightarrow$ A = A${}^2$ [. AB = A]
Again BA = B $\Rightarrow$ (BA) B = B${}^2$
$\Rightarrow$B (AB) = B${}^2$ $\Rightarrow$ B (A) = B${}^2$ [$\because$ AB = A]
$\Rightarrow$BA = B${}^2$ $\Rightarrow$ B = B${}^2$ [$\because$ BA = B]
Thus A${}^2$ = A, B${}^2$ = B
$\therefore$ A and B are idempotent matrices.