If A and B are two non-zero n × n matrices such that A 2 + B = A 2 B , then [2023]

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Solution

Q.
If A and B are two non-zero $n \times n$ matrices such that $A^{2} + B = A^{2} B,$ then [2023]

1 $A^{2} = I$ or $B = I$

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

3 $A B = I$

4 $A^{2} B = I$

Ans.
(2)

$A^{2} + B = A^{2} B \Rightarrow A^{2} + B - A^{2} B = 0$

$\Rightarrow A^{2} - A^{2} B + B = 0 \Rightarrow A^{2} (I - B) + B = 0$

$\Rightarrow A^{2} (I - B) + B - I = - I \Rightarrow (A^{2} - I) (I - B) = - I$

$\Rightarrow (A^{2} - I) (B - I) = I \Rightarrow A^{2} B - A^{2} - B = 0$

$\Rightarrow A^{2} (B - I) = B \Rightarrow A^{2} = B {(B - I)}^{- 1} \Rightarrow A^{2} = B (A^{2} - I)$

$\Rightarrow A^{2} = B A^{2} - B \Rightarrow A^{2} + B = B A^{2} \Rightarrow A^{2} B = B A^{2}$

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