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exists a 3x3 matrix with constant sums and constant products also ?

Source: Spanish Mathematical Olympiad 1985 P8

August 2, 2018
linear algebramatrixmagic squarecombinatorics

Problem Statement

A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant. Determine if there exist 3×33\times 3 matrices of real numbers which are both sum-magic and product-magic.