Question

Let $A$ and $B$ be two matrices different from $I$ such that $AB=BA$ and $A^n-B^n$ is invertible for some positive integer $n$. If $A^n-B^n=A^{n+1}-B^{n+1}=A^{n+2}-B^{n+2}$, then

• A. $I-A$ is singular
• B. $I-B$ is singular
• C. $A+B=AB+I$
• D. $(I-A)(I-B)$ is non-singular

Answer 1

Answer: (A), (B), (C)

The correct options are
A $I-A$ is singular
B $I-B$ is singular
C $A+B=AB+I$

$A^{n+2}-B^{n+2}=(A+B)(A^{n+1}-B^{n+1})-AB(A^n-B^n)$
$\Rightarrow A^n-B^n=(A+B)(A^{n-1}-B^{n-1})-AB(A^{-1}n-B^{n-1})$
$\Rightarrow I=A+B-AB[\because A^n-B^{n}isinvertible]$
$\Rightarrow (I-A)(I-B)=0$
As $A,B\ne I$, we get
$I-A$ and $I-B$ are singular matrices.
Hence, options A, B and C.