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  4. If AB = A and BA = B, then B2 is equal to

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Q. If $AB = A$ and $BA = B$, then $B^2$ is equal to

Matrices

Solution:

Since $BA = B \therefore (BA) B = (B) (B) = B^2$
$\Rightarrow B (AB) = B^2\:\: \Rightarrow BA = B^2$ $[\because AB = A] $
$ \Rightarrow B = B^2 \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, [\because BA = B] $