If \(A = \begin{bmatrix} x_1 & x_2 & 7 \\ y_1 & y_2 & y_3 \\ z_1 & 8 & 3 \end{bmatrix}\) is a matrix such that the sum of all three elements along any row, column or diagonal are equal to each…

The determinant is 288. If the centre entry is $m$ and the common line sum is $S$, summing the four lines through the centre gives $4S=3S+3m$, so $S=3m$ and every pair opposite across the centre sums to $2m$. The bottom row therefore gives $(2m-7)+8+3=3m$, so $m=4$, and completing the rows gives $\begin{bmatrix}5&0&7\\6&4&2\\1&8&3\end{bmatrix}$. Expansion across the top row gives $5(12-16)+7(48-4)=288$, while 328 comes from making the first term positive.