(1) Symmetric matrix : A square matrix \[A=[{{a}_{ij}}]\]is called symmetric matrix if \[{{a}_{ij}}={{a}_{ji}}\]for all i, j or \[{{A}^{T}}=A\].
Example : \[\left[ \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{matrix} \right]\]
(2) Skew-symmetric matrix : A square matrix \[A=[{{a}_{ij}}]\]is called skew- symmetric matrix if \[{{a}_{ij}}=-{{a}_{ji}}\]for all i, j or \[{{A}^{T}}=-A\].
Example : \[\left[ \begin{matrix} 0 & h & g \\ -h & 0 & f \\ -g & -f & 0 \\ \end{matrix} \right]\]
All principal diagonal elements of a skew- symmetric matrix are always zero because for any diagonal element.
\[{{a}_{ij}}=-{{a}_{ij}}\Rightarrow {{a}_{ij}}=0\]
Properties of symmetric and skew-symmetric matrices
(i) If A is a square matrix, then \[A+{{A}^{T}},A{{A}^{T}},{{A}^{T}}A\] are symmetric matrices, while \[A-{{A}^{T}}\]is skew- symmetric matrix.
(ii) If A is a symmetric matrix, then\[-A,KA,{{A}^{T}},{{A}^{n}},{{A}^{-1}},{{B}^{T}}AB\] are also symmetric matrices, where \[n\in N\],
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