mx = {{a, b, c, d}, {e, f, g, h}, {i, j, k, l}, {m, n, o, p}};
mx2 = {{a, a + b, a + b + c, a + b + c + d}, {e, e + f, e + f + g, e + f + g + h},
{i, i + j, i + j + k, i + j + k + l}, {m, m + n, m + n + o, m + n + o + p}};
A variation using FoldList
:
FoldList[Apply[Plus]@*Flatten@*List, Nothing, #] & /@ mx === mx2
(*True*)