Question

If A and B are symmetric and commute, then which of the following is/are symmetric?

• A. $A^{-1}B$
• B. $A{B}^{-1}$
• C. $A^{-1}{B}^{-1}$
• D. none of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A $A^{-1}B$
B $A{B}^{-1}$
C $A^{-1}{B}^{-1}$

$AB=BA(\because A$ & $B$ are commute)

$A^T=A$ & $B^T=B(\because A,B$ are symmetric)

Now $(A^{-1}B{)}^T=B^T.(A^{-1}{)}^T=B{A}^{-1}(\because$ If $A$ is symmetric then $A^{-1}$ is also symmetric)

$\therefore$ Option $A$ is true

$AB=BA$

$\therefore AB.A^{-1}=BA.A^{-1}$ (Post multiply with $A^{-1}$)

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

Now $A^{-1}AB{A}^{-1}=A^{-1}B$ (Pre multiply with $A^{-1}$)

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

$\therefore (A^{-1}B{)}^T=A^{-1}B$ (If $A^{-1}B$ is symmetric, then $B^{-1}A$ is also symmetric)

Now

$BA=AB$

$\Rightarrow (BA{)}^{-1}=(AB{)}^{-1}$ (Taking inverse)

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

Taking transpose

$(A^{-1}{B}^{-1}{)}^T=(B^{-1}{A}^{-1}{)}^T$

$=\left(A^{-1}{)}^T\right(B^{-1}{)}^T$

$=A^{-1}{B}^{-1}$

$\therefore$ Option $A,B,C$ are correct