Question

Let $A$ and $B$ be two matrices different from $I$ such that $AB=BA$ and $A^n–B^n$ is invertible for some natural number $n$. If $A^n–B^n=A^n+1–B^n+1=A^{n+2}–B^{n+2}$, then
I. $(I–A)$ is singular
II. $(I–B)$ is singular
III. $A+B=AB+I$
IV. $(I–A)(I–B)$ is non-singular

• A. I, II, III, IV
• B. I, III, IV only
• C. I, II, III only
• D. I, II, IV only

Answer 1

The correct option is D I, II, IV only

Since matrices $A$ and $B$ are commutative hence I, II, III are true.