Question

If $A$ and $B$ are symmetric matrices and $AB=BA$, then $A^{-1}B$ is a

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

Answer 1

The correct option is A Symmetric matrix

We have $AB=BA=B^{\prime }{A}^{\prime }={\left(AB\right)}^{\prime }$

$\Rightarrow AB$ is symmetric.

Also, $AB{A}^{-1}=BA{A}^{-1}=B(\because AB=BA)$

$\Rightarrow A^{-1}AB{A}^{-1}=A^{-1}B\Rightarrow B{A}^{-1}=A^{-1}B$

Therefore, ${\left(A^{-1}B\right)}^{\prime }=\left(B{A}^{-1}\right)={\left(A^{-1}\right)}^{\prime }{B}^{\prime }=A^{\prime }B$ $[\because A^{-1}$ and $B$ are symmetric $]$

Thus, the matrix $A^{-1}B$ is symmetric.