=xyzahghbfgfcxyz =ax+hy+gzhx+by+fzgx+fy+czxyz =ax2+hxy+gz2+hxy+by2+fyz+gxz+fyz+cz2 =ax2+2hxy+by2+cz2+2gxz+2fyz

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If $[\begin{matrix} x & y & z \end{matrix}] [\begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] =$

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$[{ax}^{2} + {by}^{2} + {cz}^{2} + 2 hxy + 2 gxz + 2 fyz]$

$[{ax}^{2} + {by}^{2} + {cz}^{2} + hxy + gxz + fyz]$

$[2 {ax}^{2} + 2 {by}^{2} + 2 {cz}^{2} + 2 hxy + 2 gxz + 2 fyz]$

$[2 {ax}^{2} + 2 {by}^{2} + 2 {cz}^{2} + hxy + gxz + fyz]$

answer is A.

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Detailed Solution

$= [\begin{matrix} x & y & z \end{matrix}] [\begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} \begin{matrix} a x + h y + g z & h x + b y + f z & g x + f y + c z \end{matrix} \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} \begin{matrix} a x^{2} + h x y + g z^{2} + h x y + b y^{2} + f y z + g x z + f y z + c z^{2} \end{matrix} \end{matrix}] = [\begin{matrix} \begin{matrix} a x^{2} + 2 h x y + b y^{2} + c z^{2} + 2 g x z + 2 f y z \end{matrix} \end{matrix}]$