If $A,B$ and $AB$ are symmetric matrices then is $A^{-1}B^{-1}$ also symmetric.

Question

if $ A,B$ and$ AB$ are symmetric matrices then is $A^{-1}B^{-1}$ also symmetric ?

my approach: since $$AB=BA$$, pre multiply by $A^{-1}$ and then post multiply by $A^{-1}$ to get

$$BA^{-1}=A^{-1}B$$

hence I was able to prove that $BA^{-1} and A^{-1}B $ are symmetric.(taking transpose and applying reversal property)

but using this nothing could be concluded for $A^{-1}B^{-1}$

so is there any way to prove or disprove that $A^{-1}B^{-1}$ is symmetric? thanks for your help.

Answer 1

$$\begin{align}(A^{-1}B^{-1})^T &= (B^{-1})^T(A^{-1})^T\\ &= (B^{T})^{-1}(A^{T})^{-1}\\ &= (B)^{-1}(A)^{-1}\\ &= B^{-1}A^{-1}\\ &= (AB)^{-1}\\ &= (BA)^{-1}\\ &= A^{-1}B^{-1} \end{align}$$

Answer 2

(A^-1B^-1)^T = (B^-1)^T(A^-1)^T = (B^T)^-1(A^T)^-1 = (B^-1)(A^-1) =(AB)^-1 = (BA)^-1 = A^-1.B^-1