1. Tardigrade
  2. Question
  3. Mathematics
  4. If A and B are symmetric non-singular matrices such that A B=B A and A-1 B-1 exists then matrix A-1 B-1 is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $A$ and $B$ are symmetric non-singular matrices such that $A B=B A$ and $A^{-1} B^{-1}$ exists then matrix $A^{-1} B^{-1}$ is

Matrices

Solution:

$\left(A^{-1}\right.\left.B^{-1}\right)^{T} $
$=\left(B^{-1}\right)^{T}\left(A^{-1}\right)^{T} $
$=\left(B^{T}\right)^{-1}\left(A^{T}\right)^{-1} $
$=B^{-1} A^{-1}$
$=(A B)^{-1} $
$=(B A)^{-1}=A^{-1} B^{-1} $
Hence $A^{-1} B^{-1}$ is symmetric.