Concept:

Symmetric matrix is a square matrix that is equal to its transpose.

i.e. A = AT.

A skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative.

i.e.  AT = - A

(A + B)T = (A)T  + (B)T 

(AT)T  = A

Calculations:

Given, A is any square matrix and AT is transpose matrix A

Consider the statement "A + AT is always symmetric."

We know that "a symmetric matrix is a square matrix that is equal to its transpose."

i.e. A = AT.

Consider, the matrix A + AT ....(1)

Taking transpose of the matrix A + AT 

(A + AT)^T = (A)^T  + (AT)T 

⇒(A + AT)^T = (A)T + A

⇒(A + AT)T = A + (A)T 

⇒ A + AT is symmetric.

Hence, the statement "A + AT is always symmetric is true.

Consider the statement "A - AT is always anti-symmetric" 

A skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative.

i.e.  AT = - A

Consider, the matrix A - AT ....(1)

Taking transpose of the matrix A - AT 

(A - AT)T = (A)T  - (AT)T 

⇒(A - AT)T = (A)T - A

⇒(A - AT)T = - (A - AT)

⇒ A - AT is symmetric.

Hence, the statement "A - AT is always symmetric is true.

Hence, the following statements in respect of a square matrix A and its transpose AT :

1. A + AT is always symmetric.

2. A - AT is always anti-symmetric.

both are correct