Question

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

• A. symmetric
• B. skew-symmetric
• C. Identity
• D. none of these

Answer 1

Answer: (A)

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