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

Answer 1

(b) A is a zero matrix

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

If $A=\left[a_{ij}\right]$ is a symmetric matrix, then
$a_{ij}=a_{ji}$ for all i, j ...(1)

If $A=\left[a_{ij}\right]$ is a skew-symmetric matrix, then
$a_{ij}=-a_{ji}$ for all i, j
$\Rightarrow a_{ji}=-a_{ij}$ for all i,j ...(2)

From eqs. (1) and (2), we have

$$\begin{gathered} a_{ij}=-a_{ij} \\ \Rightarrow a_{ij}+a_{ij}=0 \\ \Rightarrow 2a_{ij}=0 \\ \Rightarrow a_{ij}=0 \\ \therefore A=\left[a_{ij}\right]\mathrm{is}\mathrm{a}\mathrm{zero}\mathrm{matrix}\mathrm{or}\mathrm{null}\mathrm{matrix}. \end{gathered}$$