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  4. If A and B are two non-zero n × n matrics such that A 2+ B = A 2 B, then

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Q. If $A$ and $B$ are two non-zero $n \times n$ matrics such that $A ^2+ B = A ^2 B$, then

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Solution:

$ A ^2+ B = A ^2 B $
$ \left( A ^2- I \right)( B - I )= I $.......(1)
$ A ^2+ B = A ^2 B$
$A ^2( B - I )= B $
$A ^2= B ( B - I )^{-1} $
$ A ^2= B \left( A ^2- I \right) $
$ A ^2= BA A ^2- B $
$ A ^2+ B = BA ^2$
$ A ^2 B = BA ^2$