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## Proof That the Set of Semi Magic Squares Forms a Vector Space

Semi magic squares are square matrices suxch that the rows and columns all add up to the same number.
Example:
$\left( \begin{array}{ccc} 1 & 8 & 6 \\ 9 & 4 & 2 \\ 5 & 3 & 7 \end{array} \right)$

is a semi magic square since all the rows add up to 15, as do all the columns.
The set of semi magic squares of any particular order form a vector space
$\mathbf{V}$
.
1. The zero matrix
$\mathbf{0} \in \mathbf{V}$
since all the rows add to zero, as do all the columns.
2. Suppose all the rows/columns of the semi magic square
$\mathbf{A}$
$a$
$\mathbf{B}$
$b$
$m \mathbf{A} + n \mathbf{B}$
$ma+nb$