If A is a symmetric and B skew symmetric matrix and `(A+ B)` is non-singular and `C = (A+B)^-1 (A-B)`, then prove that

Options

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

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^(T) (A+B) ^(T)]C` [from Eq. (ii)]
`= [(A+ B)C]^(T) C`
`= (A - B) C` [from Eq. (i)]
`=(A + B) C` [from Eq. (iii) ]
` = A - B ` [ffrom Eq. (i)]