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  4. If AB = A, BA = B for any two square matrices A and B, then A2 is equal to

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Q. If $AB = A, BA = B$ for any two square matrices $A$ and $B$, then $ {{A}^{2}} $ is equal to

Rajasthan PETRajasthan PET 2003

Solution:

Given, $ AB=A\Rightarrow B=\text{ }I $
and $ BA=B\Rightarrow A=I $
Now, $ {{A}^{2}}=A.A $ $ =AB.A $
$ =A.BA $ $ =I.B $
$ =B.B={{B}^{2}} $