The numbers, in order, of each row and the numbers, in order, of each column of a <img alt="$5 \times 5$" height="12" src="https://latex.artofproblemsolving.com/1/9/7/197151a067f49bd4daa1952d1a4fac322409f0a6.png" width="40" /> array of integers form an arithmetic progression of length <img alt="$5{.}$" height="14" src="https://latex.artofproblemsolving.com/a/7/0/a70b0e7b72772204574d6d87ea5d4de63fb0bc63.png" width="16" />The numbers in positions <img alt="$(5, 5), \,(2,4),\,(4,3),$" height="20" src="https://latex.artofproblemsolving.com/c/9/4/c94e1fff1d38b5c2d4411ef89962dac0ee2a0208.png" width="150" /> and <img alt="$(3, 1)$" height="18" src="https://latex.artofproblemsolving.com/d/2/6/d261236ea904095d8d185a456444545b5206013b.png" width="38" /> are <img alt="$0, 48, 16,$" height="17" src="https://latex.artofproblemsolving.com/2/4/9/24932729499748b63460b060c3a38fbce88f03cb.png" width="68" /> and <img alt="$12{,}$" height="17" src="https://latex.artofproblemsolving.com/7/4/c/74c9c67043392ff12de73dd7c444ee5fa110fe3e.png" width="25" /> respectively. What number is in position <img alt="$(1, 2)?$" height="20" src="https://latex.artofproblemsolving.com/b/3/d/b3dc9c4c68d7465ac45b311fa56909c2a6c922be.png" width="51" /> <img alt="\[\begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]" height="96" src="https://latex.artofproblemsolving.com/2/c/a/2caa51c2881f478a1e28bf45df93c383dff268f6.png" width="151" />

Question No: 345

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 $(1, 2)?$

$\begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}$

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Tags: Algebra

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