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.

 

[CellCoree]

of a response

Cramer's rule is a method for solving systems of linear equations by using determinants. It involves replacing one column of the coefficient matrix with the constants from the right-hand side of the equations and then calculating the determinant of the resulting matrix. However, this method cannot be applied by replacing rows instead of columns.

This is because Cramer's rule relies on the properties of determinants, which are not symmetric when it comes to rows and columns. When we replace a column with the constants, we are essentially creating a new matrix with the same number of rows and columns as the original matrix. This is necessary for calculating the determinant, as it is only defined for square matrices.

On the other hand, if we were to replace a row with the constants, we would end up with a matrix that has a different number of rows and columns, making it impossible to calculate the determinant. Furthermore, the properties of determinants do not hold when we replace rows instead of columns, making the resulting solution invalid.

In short, the reason why Cramer's rule is not 'symmetric' is due to the properties of determinants and the requirement for square matrices. While we can use elementary transformations to find inverses by replacing rows or columns, this method does not apply to solving non-homogeneous linear equations using Cramer's rule.