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. Unit matrix
• D. Column Matrix

Answer 1

The correct option is A

Symmetric matrix

AB = BA = B 'A' = (AB)' ⇒ AB is symmetric
Also, $AB{A}^{-1}=BA{A}^{-1}$= B
$\Rightarrow A^{-1}AB{A}^{-1}=A^{-1}B\Rightarrow B{A}^{-1}=A^{-1}B$
$\Rightarrow (A^{-1}B{)}^{\prime }=(B{A}^{-1}{)}^{\prime }=(A^{-1}{)}^{\prime }{B}^{\prime }=A^{-1}B$

$(A^{-1}{)}^{\prime }{B}^{\prime }=A^{-1}B$. Therefore $A^{-1}B$ is symmetric.