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

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

Matrices

Solution:

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.