Taking inverse Fourier transform in column-wise and solve it in row-wise
No. Your problem doesn't put any constraints on the columns, so they're totally independent. Then, transforming them to a different base doesn't change that, at all.

If the entries in different columns had nothing to do with other columns, that won't change. The fact that you arranged your data in a matrix doesn't magically allow any operation to introduce relationships that weren't there.