Q.If $A$ is a square matrix such that $A^T=A$, then $A$ is called:

A matrix that is unchanged by transposing has a specific standard name distinct from the other options.

Testing each option against the definitions

1. A skew-symmetric matrix is defined by $A^T=-A$, not $A^T=A$ — the opposite sign condition, so (a) is wrong.
2. An orthogonal matrix is defined by $A^TA=I$ (its transpose equals its inverse) — a different condition entirely, not covered in this chapter and not implied merely by $A^T=A$.
3. A symmetric matrix is defined precisely by $A^T=A$ — every entry $a_{ij}$ equals $a_{ji}$. This matches the question exactly, so (c) is correct. …

Assuming a symmetric matrix must also be diagonal (it need not be — only its off-diagonal entries must come in matching PAIRS, not be zero), and mi …