Question

Let $A$ and $B$ be two $n\times n$ matrices such that $det\left(A\right)\ne 0,A+B=(AB{)}^2$ and $BAB=A+I$, then
$\left(I\right)A^{-1}=(A^4-I)\left(II\right)B^5-A^5=A+B\left(III\right)A^9=A^4+A+I\left(IV\right)A^2{B}^2=B{A}^{2}B$

Which of the following is correct?
​​​​​​​(correct answer + 1, wrong answer - 0.25)

• A. Only $\left(I\right)$ is true
• B. Only $\left(I\right)$ and $\left(IV\right)$ are true
• C. Only $\left(I\right),\left(III\right)$ and $\left(IV\right)$ are true
• D. All are true

Answer 1

The correct option is D All are true

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

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

$B^5-A^5=A^{10}-A^5=(A+I{)}^2-(A+I)=A^2+A=A+B$

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

$A^2{B}^2=A^6=A^2{A}^2{A}^2=B{A}^{2}B$