Question

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

• A. True
• B. False

Answer 1

The correct option is A True

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

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

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

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

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

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