matrix is both symmetric and skew-symmetric matrix.

Let $A=\left[a_{i j}\right]$ be a matrix which is both symmetric and skew-symmetric.\

$A$ is a symmetric matrix.

$\therefore a_{i j}=a_{j i}$, for all $i, j$      ....(1)

Also, $A$ is a skew-symmetric matrix.

$\therefore a_{i j}=-a_{j i}$, for all $i, j$     .....(2)

From (1) and (2), we have

$a_{i j}=-a_{i j}$, for all $i, j$

$\Rightarrow 2 a_{i j}=0$, for all $i, j$

$\Rightarrow a_{i j}=0$, for all $i, j$

$\therefore$ The matrix $A$ is a : matrix.

: matrix is both symmetric and skew-symmetric matrix