Write a square matrix which is both symmetric as well as skew-sym… - Vidyadip

Question

Write a square matrix which is both symmetric as well as skew-symmetric.

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Answer

Let $\text{A}=\begin{bmatrix}0&0\\0&0 \end{bmatrix}$
$\text{A}^{\text{T}{}}=\begin{bmatrix}0&0\\0&0 \end{bmatrix}$
Since $A^T = A, A$ is a symmetric matrix.
Now,
$-\text{A}=-\begin{bmatrix}0&0\\0&0 \end{bmatrix}$
$\Rightarrow-\text{A}=\begin{bmatrix}0&0\\0&0 \end{bmatrix}$
Since $A^T = -A, A$ is a skew-symmetric matrix.
Thus, $\text{A}=\begin{bmatrix}0&0\\0&0 \end{bmatrix}$is an exampal of a matrix that is both symmetric and skew-symmetric.

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