Step-by-Step Solution
Key Concept: For symmetric matrices, $A + A^T = 2A$, and diagonal elements can be determined from trace conditions
Since $A + A^T$ is symmetric and $a_{ij} + a_{ji} = 6$ for all $i≠j$, we have $AM = GM$ with $a_{12} + a_{21} = 6$, $a_{13} + a_{31} = 6$, $a_{23} + a_{32} = 6$. For diagonal elements: $2a_{11} = 10$ so $a_{11} = 5$; $2a_{22} = 6$ so $a_{22} = 3$; $2a_{33} = 8$ so $a_{33} = 4$. With $A = \begin{pmatrix} 5 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 4 \end{pmatrix}$, we get $|A| = 5(12-1) - 2(8-3) + 3(2-9) = 5(11) - 2(5) + 3(-7) = 55 - 10 - 21 = -37$.
Correct Answer: -37