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.