Question

If $A$ is a symmetric and $B$ is skew-symmetric matrix and $(A+B)$ is non-singular and $C=(A+B{)}^{-1}(A-B)$, then which of the following options is/are correct

• A. $C^T(A+B)C=A+B$
• B. $C^T(A+B)C=A-B$
• C. $C^T(A-B)C=A-B$
• D. $C^T(A-B)C=A+B$

Answer 1

Answer: (A), (C)

$C^T(A-B)C=A-B$

$A$ is a symmetric matrix
$\Rightarrow A^T=A$
$B$ is a skew symmetric matrix.
$\Rightarrow B^T=-B$
Now,
$(A+B)C=(A+B)(A+B{)}^{-1}(A-B)$

$\Rightarrow (A+B)C=(A-B)\cdots \left(1\right)$
$C^T=(A-B{)}^T((A+B{)}^{-1}{)}^T$
$=(A+B)\left(\right(A+B{)}^T{)}^{-1}$
$=(A+B)(A-B{)}^{-1}\cdots (2)$

where,$|A+B|\ne 0\Rightarrow |(A+B{)}^T|\ne 0\Rightarrow |A-B|\ne 0$

From $\left(1\right)$ and $\left(2\right)$ we have:
$C^T(A+B)C=(A+B)(A-B{)}^{-1}(A-B)$
$C^T(A+B)C=(A+B)\cdots \left(3\right)$
Taking transpose in $\left(3\right)$
$C^T(A+B{)}^T(C^T{)}^T=(A+B{)}^T$
$C^T(A-B)C=A-B$