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

If A and B are two non-zero n×n matrices such that A2+B=A2B, then
  1. AB=I
  2. A2B=I
  3. A2=I or B=I
  4. A2B=BA2

Solution

Given:

A2B=A2+B

⇒A2B-A2-B+I=I

⇒A2B-I-B-I=I

⇒A2-IB-I=I

⇒A2-IB-IB-I-1=IB-I-1

⇒A2-I=B-I-1

Also,

A2+B=A2B

⇒A2B-I=B

⇒A2=BB-I-1

⇒A2=BA2-I

⇒A2=BA2-B

⇒A2+B=BA2

⇒A2B=BA2

Asked in: JEE Main 2023 (24 Jan Shift 1)

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