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Singular matrices applied to 3 × 3 magic squares

  • N. Gauthier (a1)


If you enjoy magic squares and their unusual and fascinating numerical properties, you are in good company. Indeed, many a professional mathematician, scientist and amateur has shared the same interest. Technical research work on the properties of magic squares is available aplenty in the literature or in books concerned with the recreational aspects of mathematics. Studies of magic squares naturally lead to some elements of group theory, of lattices, of Latin squares, of partitions, of matrices, of determinants, and so on …

By definition, the general n x n magic square is a square array of n 2 numbers {aij ; i, j = 1,2,... , n) whose n rows, n columns, and two main diagonals have the same sum s. It is therefore natural to think of a magic square as a matrix and we shall do this in the present note, simply referring to it as a magic matrix.



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1. Trainor, G., Ross, E., Doherty, A., Livingstone, M. and Nicholson, A., A Further Mathematics investigation: A × B = B, Math. Gaz. 70 (October 1986) pp. 192197.
2. MacNeill, J., Some properties of powers of 3 × 3 matrices, Math. Gaz. 72 (June 1988) pp. 8992.
3. Brakes, W.R., Unexpected groups, Math. Gaz. 79 (November 1995) pp. 513520.
4. Eperson, D.B., Properties of the general 3 by 3 magic square, Math. Gaz. 79 (July 1995) pp. 382383.
5. Thompson, A.C., Odd magic powers, Amer. Math. Monthly 101(4) (April 1994) pp. 339342.
6. Hill, R. and Elzaidi, S., Cubes and inverses of magic squares, Math. Gaz. 80 (November 1996) pp. 565567.

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Singular matrices applied to 3 × 3 magic squares

  • N. Gauthier (a1)


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