Use Total with the appropriate second argument to sum the matrix along rows/columns.
###Sum along rows:
Sum along rows:
m = {{a,b,c}, {d,e,f},{g,h,i}}; Total[m, {1}] (* {a + d + g, b + e + h, c + f + i} *) By default, Total[m] (without a second argument) sums along the rows.
###Sum along columns
Sum along columns
Total[m, {2}] (* {a + b + c, d + e + f, g + h + i} *)