Question

If A is a symmetric and B skew-symmetric matrix and $A+B$ is non-singular and $C=(A+B{)}^{-1}(A-B),$ then find $C^T(A-B)C.$

• A. $(A-B{)}^2$
• B. $A+B$
• C. $A-B$
• D. None of these.

Answer 1

Answer: (C)

$A-B$

$C=(A+B{)}^{-1}(A-B),$

$\Rightarrow (A+B)C=(A+B)(A+B{)}^{-1}(A-B)$

$\Rightarrow (A+B)C=A-B$ (1)

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

$=(A+B)\left(\right(A+B{)}^T{)}^{-1}$ as A is symmetric and B in anti-symmetric matrices.

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

$=(A+B)(A-B{)}^{-1}$ (2)

From (1) and (2), we get

$C^T(A+B)C\ne (A+B)(A-B{)}^{-1}(A-B)$

$=(A+B)$ (3)

Taking transpose in (3), we get

$C^T(A+B{)}^T(C^T{)}^T=(A+B{)}^T$

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