If A and B are two non-zero n × n matrices such that A²+B=A²B, then :

Solution

Maths sir · 4 steps

Add I to both sides and factorise: two factors whose product is I must commute.

1. Rearrange and add I. Move everything to one side: A²B - A² - B = O. Add I to both sides to get A²B - A² - B + I = I.
2. Factorise. Expand (A² - I)(B - I) = A²B - A² - B + I. This is exactly the left side, so the condition becomes a product equal to I.

   (A² - I)(B - I) = I

3. Inverses commute. So A² - I and B - I are inverses of each other, and a matrix commutes with its inverse. That gives (B - I)(A² - I) = I, which expands to BA² - B - A² + I = I, so BA² = A² + B.
4. So. Both A²B and BA² equal A² + B, so A²B = BA². The answer is (c).

The idea to keep. When you see XY = X + Y with matrices, add I and write (X - I)(Y - I) = I. Inverses always commute, so XY = YX follows at once.

Answer from the NTA final answer key. Solution by PrepWiser.