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

Question

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

MARKS App · Accepted Answer

Given: A 2 B = A 2 + B ⇒ A 2 B - A 2 - B + I = I ⇒ A 2 B - I - B - I = I ⇒ A 2 - I B - I = I ⇒ A 2 - I B - I B - I - 1 = I B - I - 1 ⇒ A 2 - I = B - I - 1 Also, 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 2 - B ⇒ A 2 + B = B A 2 ⇒ A 2 B = B A 2

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