Express the following matrices as the sum of a symmetric and a skew symmetric matrix :
$⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦$

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Solution

$\begin{matrix} A = \frac{1}{2} (A + A^{^{'}}) + \frac{1}{2} (A - A^{^{'}}) A = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ A^{^{'}} = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ A = \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 12 & - 4 & 4 - 4 & 6 & - 2 4 & - 2 & 6 \end{matrix} ⎤ ⎥ ⎦ + \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦ A = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ a s A = A^{^{'}} ∴ A - A^{^{'}} = 0 \end{matrix}$

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