If a matrix A is both symmetric and skew-symmetric, then
Question:

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

Solution:

(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 null matrix.