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

Question

If A and B are two non-zero n × n matrics such that A 2+ B = A 2 B, then (A) A 2= I or B = I (B) A 2 B = BA 2 (C) AB = I (D) A 2 B = I

Tardigrade Admin · Accepted Answer

A 2+ B = A 2 B ( A 2- I )( B - I )= I .......(1) A 2+ B = A 2 B A 2( B - I )= B A 2= B ( B - I )-1 A 2= B ( A 2- I ) A 2= BA A 2- B A 2+ B = BA 2 A 2 B = BA 2

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

$A^{2} = I$ or $B = I$

$A^{2} B = B A^{2}$

$A B = I$

$A^{2} B = I$

$A^{2} + B = A^{2} B$
$(A^{2} - I) (B - I) = I$ .......(1)
$A^{2} + B = A^{2} B$
$A^{2} (B - I) = B$
$A^{2} = B (B - I)^{- 1}$
$A^{2} = B (A^{2} - I)$
$A^{2} = B A A^{2} - B$
$A^{2} + B = B A^{2}$
$A^{2} B = B A^{2}$

Q. If A and B are two non-zero n×n matrics such that A2+B=A2B, then

A2=I or B=I

A2B=BA2

AB=I

A2B=I