Question

If A and B are two non singular matrices and both are symmetric and commute each other then

• A. Both $A^{-1}B$ and $A^{-1}{B}^{-1}$ are symmetric
• B. $A^{-1}B$ is symmetric but $A^{-1}{B}^{-1}$ is not symmetric
• C. $A^{-1}{B}^{-1}$ is symmetric but $A^{-1}B$ is not symmetric
• D. Neither $A^{-1}B$ nor $A^{-1}{B}^{-1}$ are symmetric

Answer 1

The correct option is A Both $A^{-1}B$ and $A^{-1}{B}^{-1}$ are symmetric

AB =BA
Previous & past multiplying both sides by $A^{-1}$.
$A^{-1}\left(AB\right)A^{-1}=A^{-1}\left(BA\right)A^{-1}\left(A^{-1}A\right)\left(B{A}^{-1}\right)=A^{-1}B\left(A{A}^{-1}\right)\Rightarrow (B{A}^{-1}{)}^1=(A^{-1}B{)}^1=\left(A^{-1}{)}^1{B}^1\right(reversallaws)=A^{-1}B(asB=B^1\left)\right(A^{-1}{)}^1=A^{-1}\Rightarrow A^{-1}B$ is symmetric
Similarly for $A^{-1}{B}^{-1}$.