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Broken diagonal

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In recreational mathematics and the theory of magic squares, a broken diagonal is a set of n cells forming two parallel diagonal lines in the square. Alternatively, these two lines can be thought of as wrapping around the boundaries of the square to form a single sequence.

In pandiagonal magic squares

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A magic square in which the broken diagonals have the same sum as the rows, columns, and diagonals is called a pandiagonal magic square.[1][2]

Examples of broken diagonals from the number square in the image are as follows: 3,12,14,5; 10,1,7,16; 10,13,7,4; 15,8,2,9; 15,12,2,5; and 6,13,11,4.

The fact that this square is a pandiagonal magic square can be verified by checking that all of its broken diagonals add up to the same constant:

3+12+14+5 = 34
10+1+7+16 = 34
10+13+7+4 = 34

One way to visualize a broken diagonal is to imagine a "ghost image" of the panmagic square adjacent to the original:

The set of numbers {3, 12, 14, 5} of a broken diagonal, wrapped around the original square, can be seen starting with the first square of the ghost image and moving down to the left.

In linear algebra

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Broken diagonals are used in a formula to find the determinant of 3 by 3 matrices.

For a 3 × 3 matrix A, its determinant is

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Here, and are (products of the elements of) the broken diagonals of the matrix.

Broken diagonals are used in the calculation of the determinants of all matrices of size 3 × 3 or larger. This can be shown by using the matrix's minors to calculate the determinant.

References

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  1. ^ Pickover, Clifford A. (2011), The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across the Dimensions, Princeton University Press, p. 7, ISBN 9781400841516.
  2. ^ Licks, H. E. (1921), Recreations in Mathematics, D. Van Nostrand Company, p. 42.
  3. ^ title=Determinant|url=https://mathworld.wolfram.com/Determinant.html