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Question
Write a square matrix which is both symmetric as well as skew-symmetric.
Solution
\[Let A = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} \]
\[ A^T = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
`"Since" A^T = A, A is a symmmetric matrix `
\[Now, \]
\[ - A = - \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} \]
\[ \Rightarrow - A = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
`"Since" A^T = - A, A is a skew - symmetric matrix . `
Thus,` A= [[0 0 ],[0 0]] `is an example of a matrix that is both symmetric and skew - symmetric.
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