Why is Cramer's rule for determinants not 'symmetric'?

[Jinius]

we can solve non-homogeneous equations in matrix form using Cramer's rule. This rule is valid only if we are replacing the columns. Why can't we replace the rows and carry on the same? For eg we can use elementary transformations for obtaining inverses either via rows or via columns.
But we can't find solutions to non homogeneous linear equations by replacing rows. Could someone please explain this? I am in a need

 

[HallsofIvy]

You can! Who told you you can't? Swapping rows and columns will not change a determinant.

 

[elibj123]

You can! Who told you you can't? Swapping rows and columns will not change a determinant.

Au=v

Yes, but in Cramer's rule you plug v as a column in A, so you swap certain data. If you plug v as a row in A, you will swap another data, and the determinant will certainly change.

As to the question itself, I think that's how it is. In this order (Au=v) the columns of A are the coefficients of each variable u1,2,3,... Therefore to pull data on u1 you will have to swap the first column and not the first row.

 

[HallsofIvy]

I am asuming that, by "using rows rather than columns, the OP simply meant taking the transpose. Otherwise, the question just doesn't make sense.