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JEE Main202324 Jan 2023Morning ShiftMathematicsMatricesActual

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

Options

  1. AA B = I
  2. BA 2 B = I
  3. CA 2 = I   or   B = I
  4. DA 2 B = B A 2

Correct answer

D. A 2 B = B A 2

Step-by-step solution

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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