Correct Answer - Option 3 : Both 1 and 2
Concept:
Consider a matrix A and its transpose AT:
(AT)T = A, (A + B)T = AT + BT
A matrix ‘A’ is said to be symmetric if A = AT and anti-symmetric if A = -AT
Calculation:
We know, that A matrix ‘A’ is said to be symmetric if A = AT and anti-symmetric if A = -AT
Now, consider \(\rm (A+A^T)^T=A^T+(A^T)^T =A^T+A\)
Therefore, A + AT is always symmetric.
Now, consider \(\rm (A-A^T)^T=A^T-(A^T)^T =A^T-A\)
= -(A - AT)
Therefore, A - AT is always anti-symmetric.
Hence, option (3) is correct.