Problems from Definiton and associated matrixes of analytical quadric

What is the equation associated with the matrix of the following quadric? $\overset{―}{A} = [\begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 2 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 3 & 0 & 2 & 5 \end{matrix}]$

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Development:

To find the equation of the conical curve, it is sufficient to develop the following product:

$[\begin{matrix} x & y & z & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 2 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 3 & 0 & 2 & 5 \end{matrix}] [\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}] = [\begin{matrix} x & y & z & 1 \end{matrix}] [\begin{matrix} x + z + 3 \\ 2 y \\ x + z + 2 \\ 3 x + 2 z + 5 \end{matrix}] =$ $= x^{2} + x z + 3 z + 2 y^{2} + x z + z^{2} + 2 z + 3 x + 2 z + 5 =$ $= x^{2} + 2 y^{2} + z^{2} + 2 x z + 6 x + 4 z + 5$

Solution:

The associate equation is $x^{2} + 2 y^{2} + z^{2} + 2 x z + 6 x + 4 z + 5$

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