If A is a symmetric matrix, B is a skew-symmetric matrix, `A+B` is nonsingular and `C=(A+B)^(-1) (A-B)`, then prove that
(i) `C^(T) (A+B) C=A+B` (ii) `C^(T) (A-B)C=A-B`
(iii) `C^(T)AC=A`

Options

A. `A + B`
B. `A-B`
C. A
D. B

Answer

A

Explanation

Given, `A^(T) = A, B^(T) = -B, det (A+ B) ne 0`
and `C = (A + B)^(-1) (A-B)`
`rArr (A + B) C = A - B` ...(i)
Also, ` (A + B) ^(T)= A - B` ...(ii)
and ` (A - B) ^(T)= A + B` ...(iii)
`C^(T) (A+B) C= C^(T) [(A+B)C]`
`=C^(T) (A-B)` [from Eq. (i)]
`C^(T) (A+ B)^(T)` [from Eq.(ii)]
`=[(A + B)C]^(T)`
`= (A- B)^(T)` [from Eq. (i) ]
`= A + B `[ from Eq. (iii)]