A square matrix is called half-magic
if the sum of the numbers in each row and column is the same.
The common sum in each row and column is denoted by
and is called the magic sum of the matrix .
Let be the vector space of half-magic squares.
(a) Find
.
Consider the linear operator that subtracts the magic sum from every entry in the matrix.
More preciselly, .
(b) Find
.
(c) Find an ordered basis for .
,
.
(d) Find the matrix of in your chosen basis .
.