Consider the following statements in respect of a square matrix A and its transpose `A^(T)`.
1. `A + A^(T)` is always symmetric.
2. `A-A^(T)` is always anti-symmetric
Which of the statements given above is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

C

Explanation

We know, that A matrix 'A' is said to be symmetric `A = A^(T)` and anti-symmetric if `A = -A^(T)`
Now, consider `(A+A^(T))^(T) = A^(T) + (A^(T))^(T) = A^(T) - A`
`rArr A+A^(T)` is always symmetrc
Now, consider `(A-A^(T))^(T) = A^(T) - (A^(T))^(T) = A^(T)-A=-(A-A^(T))^(T)`.
`rArr A- A^(T)` is always anti-symmetric.