If A and B are two non-zero $\mathrm{n} \times \mathrm{n}$ matrics such that $\mathrm{A}^{2}+\mathrm{B}=\mathrm{A}^{2} \mathrm{~B}$, then

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Matrices

If A and B are two non-zero

n \times n

matrics such that

A^{2} + B = A^{2} B

, then

Answer: (D)

From

A^{2} + B = A^{2} B

, bring every term to one side:

A^{2} B - A^{2} - B + I = I

, which factors as

(A^{2} - I) (B - I) = I

. Thus

A^{2} - I

and

B - I

are mutual inverses and so also satisfy

(B - I) (A^{2} - I) = I

, i.e.

B A^{2} - B - A^{2} = 0

. This gives

B A^{2} = A^{2} + B = A^{2} B

. Therefore

A^{2} B = B A^{2} .

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option