Question

Let $A$ and $B$ be two $n\times n$ matrices such that $det\left(A\right)\ne 0,$ $A+B={\left(AB\right)}^2$ and $BAB=A+I,$ where $I$ is an identity matrix. Which of the following is/are CORRECT?

• 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)

$A^2{B}^2=B{A}^{2}B$

Given that $A+B=(AB{)}^2$ and $BAB=A+I$
Now, $A+B=(AB{)}^2$
$\Rightarrow A+B=ABAB\Rightarrow A+B=A(A+I)\Rightarrow A+B=A^2+A$
$\Rightarrow B=A^2$

$BAB=A^5=A+I\Rightarrow A^4=I+A^{-1}$
$\Rightarrow A^{-1}=A^4-I$

Since $BAB=A^5=A+I,$
$\Rightarrow A^{10}=A^6+A^5$
$\Rightarrow A^{10}-A^5={\left(A^3\right)}^2$
$\Rightarrow B^5-A^5=\left(AB{)}^2\right(\because B=A^2)$
$\Rightarrow B^5-A^5=A+B(\because A+B=(AB{)}^2)$

Since $BAB=A^5=A+I,$
$\Rightarrow A^9=A^5+A^4$
$\Rightarrow A^9=A^4+A+I(\because A^5=A+I)$

$A^2{B}^2=B{A}^4=B{A}^2{A}^2=B{A}^{2}B$