Question

If $A$ and $B$ are non-singular, symmetric and commutable matrices, then $A^{-1}{B}^{-1}$ is

• A. Symmetric matrix
• B. Identity matrix
• C. Skew-symmetric matrix
• D. None of these

Answer 1

The correct option is A Symmetric matrix

Commutable $\Rightarrow AB=BA$
Symmetric $\Rightarrow A=A^T,B=B^T$
Let $C=A^{-1}{B}^{-1}$
$C^T=(A^{-1}{B}^{-1}{)}^T=(B^{-1}{)}^T(A^{-1}{)}^T$
Since, inverse of transpose is same as transpose of inverse,
$C^T=\left(B^T{)}^{-1}\right(A^T{)}^{-1}$
$\Rightarrow C^T=B^{-1}{A}^{-1}$ (both are symmetric)
$\Rightarrow C^T=(AB{)}^{-1}$
$\Rightarrow C^T=(BA{)}^{-1}$ (commutable)
$\Rightarrow C^T=A^{-1}{B}^{-1}$
$\Rightarrow C^T=C$
Hence, $C$ is a symmetric matrix.