Question

______________ matrix is both symmetric and skew-symmetric matrix.

Answer 1

Let A = [a_ij] be a matrix which is both symmetric and skew-symmetric.

A is a symmetric matrix.

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

Also, A is a skew-symmetric matrix.

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

From (1) and (2), we have

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

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

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

∴ The matrix A is a null matrix.

____Null____ matrix is both symmetric and skew-symmetric matrix.