If you view a 3 × 3 magic square as a matrix and raise it to the third power, the result is also a magic square.
More generally, if you multiply an odd number of 3 × 3 magic squares together, the result is a magic square.
For example, here are three magic squares that appeared in my blog post on Spanish magic squares, and you can verify that their product is another magic square.
![\left[ \begin{array}{ccc} 93 & 155 & 121 \\ 151 & 123 & 95 \\ 125 & 91 & 153 \\ \end{array} \right] \left[ \begin{array}{ccc} 12 & 21 & 15 \\ 19 & 16 & 13 \\ 17 & 11 & 20 \\ \end{array} \right] \left[ \begin{array}{ccc} 13 & 20 & 18 \\ 22 & 17 & 12 \\ 16 & 14 & 21 \\ \end{array} \right] = \left[ \begin{array}{ccc} 299622 & 301968 & 301722 \\ 303204 & 301104 & 299004 \\ 300486 & 300240 & 302586 \\ \end{array} \right]](https://www.johndcook.com/magic_matrix_product.svg)
Source: Martin Gardner, Some New Discoveries About 3 × 3 Magic Squares, Math Horizons, February 1998.
The same article conjectures that the results above are true for magic squares of any odd order. That was 20 years ago. Maybe the conjecture has been resolved by now. If you know, please leave a comment.
‘Powers of Magic Matrices’ by B. Edwards and J. Hartman, The Mathematical Gazette, July 2011, vol 95 #533, p284-292 may be of interest, however they do not require that the elements of the matrix are all distinct, which may not be what you are expecting.