Broken diagonal

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.

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

${\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}\Box &\Box &\Box \\\Box &e&f\\\Box &h&i\end{vmatrix}}-b\,{\begin{vmatrix}\Box &\Box &\Box \\d&\Box &f\\g&\Box &i\end{vmatrix}}+c\,{\begin{vmatrix}\Box &\Box &\Box \\d&e&\Box \\g&h&\Box \end{vmatrix}}\\[3pt]&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}$[3]

Here, ${\displaystyle bfg,cdh,bdi,}$ and ${\displaystyle afh}$ 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.