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  4. If A = [6&8&5 4&2&3 9&7&1] is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is

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Q. If $A = \begin{bmatrix}6&8&5\\ 4&2&3\\ 9&7&1\end{bmatrix} $ is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is

Matrices

Solution:

If $A = \begin{bmatrix}6&8&5\\ 4&2&3\\ 9&7&1\end{bmatrix} $ is the sum of a symmetric matrix B and skew symmetric matrix C,
Transpose of $A = \begin{bmatrix}6&4&9\\ 8&2&7\\ 5&3&1\end{bmatrix}$
So that $B = \frac{1}{2} \left[\begin{bmatrix}6&8&5\\ 4&2&3\\ 9&7&1\end{bmatrix} + \begin{bmatrix}6&4&9\\ 8&2&7\\ 5&3&1\end{bmatrix}\right] $
$B = \frac{1}{2} \begin{bmatrix}12&12&14\\ 12&4&10\\ 14&10&2\end{bmatrix} = \begin{bmatrix}6&6&7\\ 6&2&5\\ 7&5&1\end{bmatrix}$