Question

$A$ and $B$ are two $n\times n$ matrices such that $|A|\ne 0,A+B=(AB{)}^2$ and $BAB=A+I$. Choose the correct options.

• A. $A^{-1}=A^4-I$
• B. $B^5-A^5=A+B$
• C. $A^9=A^4+A+I$
• D. $A^2{B}^2=B{A}^{2}B$

Answer 1

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

The correct options are
A $A^{-1}=A^4-I$
B $B^5-A^5=A+B$
C $A^9=A^4+A+I$
D $A^2{B}^2=B{A}^{2}B$

$A+B=ABAB=A^2+A$
$\Rightarrow B=A^2$
Now,$BAB=A^5=A+I\Rightarrow A(A^4-I)=I\Rightarrow A^{-1}=A^4-I$

$B^5-A^5=(A^5{)}^2-A^5=(A+I{)}^2-(A+I)=A^2+A=A+B$

$A^4+A+I=A^4+A^5=A^4(A+I)=A^9$