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

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Solution

$L e t A = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ A^{1} = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = A$

So, $A$ is Symmetric matrix

$\frac{A + A^{1}}{2} = \frac{2 A}{2} = A$ (Symmetric matrix)

$\frac{A - A^{1}}{2} = \frac{A - A}{2} = 0$ (Skew Symmetric matrix)

therefore
$^{A = ⎛ ⎜ ⎝ \frac{A + A^{1}}{2} ⎞ ⎟ ⎠ + ⎛ ⎜ ⎝ \frac{A - A^{1}}{2} ⎞ ⎟ ⎠}$
$= A + 0$

$= ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$

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⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & - 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦

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