If a matrix $A$ is both symmetric and skew-symmetric, then
(a) $A$ is a diagonal matrix
(b) $A$ is a zero matrix
(c) $A$ is a scalar matrix
(d) $A$ is a square matrix
(b) $A$ is a zero matrix
Let $A=\left[a_{i j}\right]$ be a matrix which is both symmetric and skew-symmetric.
If $A=\left[a_{i j}\right]$ is a symmetric matrix, then
$a_{i j}=a_{j i}$ for all $\mathrm{i}, \mathrm{j}$ ....(1)
If $A=\left[a_{i j}\right]$ is a skew-symmetric matrix, then
$a_{i j}=-a_{j i}$ for all $\mathrm{i}, \mathrm{j}$
$\Rightarrow a_{j i}=-a_{i j}$ for all $\mathrm{i}, \mathrm{j}$ ....(2)
From eqs. (1) and (2), we have
$a_{i j}=-a_{i j}$
$\Rightarrow a_{i j}+a_{i j}=0$
$\Rightarrow 2 a_{i j}=0$
$\Rightarrow a_{i j}=0$
$\therefore A=\left[a_{i j}\right]$ is a zero matrix or : matrix.