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Mathematics Question on Matrices

If A=[685 423 971]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

A

[667 625 751] \begin{bmatrix}6&6&7\\\ 6&2&5\\\ 7&5&1\end{bmatrix}

B

[02−2 −25−2 220]\begin{bmatrix}0&2&-2\\\ -2&5&-2\\\ 2&2&0\end{bmatrix}

C

[667 −62−5 −751]\begin{bmatrix}6&6&7\\\ -6&2&-5\\\ -7&5&1\end{bmatrix}

D

[06−2 20−2 −2−20]\begin{bmatrix}0&6&-2\\\ 2&0&-2\\\ -2&-2&0\end{bmatrix}

Answer

[667 625 751] \begin{bmatrix}6&6&7\\\ 6&2&5\\\ 7&5&1\end{bmatrix}

Explanation

Solution

If A=[685 423 971]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=[649 827 531]A = \begin{bmatrix}6&4&9\\\ 8&2&7\\\ 5&3&1\end{bmatrix} So that B=12[[685 423 971]+[649 827 531]]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=12[121214 12410 14102]=[667 625 751]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}