------
## 最终结论
```python
def sort_matrix(matrix):
    row_sums = [sum(row) for row in matrix]
    sorted_matrix = [row for _, row in sorted(zip(row_sums, matrix))]
    return sorted_matrix

# Test cases
assert sort_matrix([[1, 2, 3], [2, 4, 5], [1, 1, 1]]) == [[1, 1, 1], [1, 2, 3], [2, 4, 5]]
assert sort_matrix([[1, 2, 3], [-2, 4, -5], [1, -1, 1]]) == [[-2, 4, -5], [1, -1, 1], [1, 2, 3]]
assert sort_matrix([[5,8,9],[6,4,3],[2,1,4]]) == [[2, 1, 4], [6, 4, 3], [5, 8, 9]]
```