2024 AMC 12A Problem 14

The numbers, in order, of each row and the numbers, in order, of each column of a 5 \times 5 array of integers form an arithmetic progression of length 5{.} The numbers in positions (5, 5), \,(2,4),\,(4,3), and (3, 1) are 0, 48, 16, and 12{,} respectively. What number is in position (1, 2)? \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix} \textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.