> Another common ways to use matrices is element-wise, where you refer to x[i][j]. Mathematicians expect i to be the row and j to be the column. Which matches C

> Another common ways to use matrices is element-wise, where you refer to x[i][j]. Mathematicians expect i to be the row and j to be the column. Which matches C. Whilst Fortran does it the other way around. That’s not quite it. The fact that the row index is first is purely a convention. And by convention, i in In X(i,j) in some Fortran code is still the row index. The difference with C is the layout in memory, not the semantics: the fact that X(i+1,j) is contiguous to X(i,j) in Fortran and not in C, which means that for efficiency reasons inner loops in Fortran are over row indices rather than column indices as in C. It would have been odd for Fortran, which has been since the beginning a language for numerical calculations, to “not match what mathematicians expect”.