Dec 132011

Question by lowbrasskicksass: How do you complete a magic square with unknowns in it with matrices and linear algebra?
I have a magic matrix with 6 unknowns:

1 a -2
b 4 c
d e f

How would you get/What would the matrix be that represents this that could be row-reduced to solve?

Thank you for any help!

Best answer:

Answer by Piyush
notice that if a matrix M is a magic square, then xM+y will also be a magic square (for a and be bee two real numbers).

the 3×3 magic square is
4 9 2
3 5 7
8 1 6
with all it’s possible permutations (actually 8, 4 if you rotate the square by 90 degrees, and if you flip the matrix, 4 more)

in all these, 5 will always be in middle position, so 4x+y=5
similarly try making equations (x+y= sth and -2x+y= sth) for all these 8 cases, and the one which gives you a unique value of x and y is solution
I hope it helps :D

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