UPSC NDA Math Model Paper 10

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 A :1. A+AT is always symmetric.
2. A−AT is always anti-symmetric.
both are correct