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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  
\[\mathbf{A}\]
  add up to  
\[a\]
  and all the rows columns of the semi magic square  
\[\mathbf{B}\]
  add up to  
\[b\]
. Then all the rows and columns of  
\[m \mathbf{A} + n \mathbf{B}\]
  add up to  
\[ma+nb\]

Hence the set of semi magic squares forms a vector space.