Determinant Question: Understanding Row Interchange & Cycles

[Weather Freak]

Hi Folks,

I have a question about determinants that is probably quite simple. I know that if you have a matrix and you interchange rows, the determinant changes. However, if you cyclically change the rows up or down, still in order, the determinant does not change.

What is the theorem or other rule that governs the difference between the two? Is it a fundamental property of matrices that perhaps I've missed along the way? I've searched through a variety of textbooks and websites, and seen that this is indeed true, but no one has provided an explanation as to why.

Thanks!

 

[radou]

It simply follows from that very rule, i.e. det B = - det A, if one obtains B by interchanging two rows or columns from A. And the rule itself follows from the definition of the determinant.

 

[dacruick]

Linear Algebra is not by any means my strong suit but if you have a 3x3 matrix and you shift the rows down, you still have the same 3 vectors that form the matrix. Wouldnt that be why the determinant doesn't change

 

[ulriksvensson]

The determinant is proportional to \epsilon_{ijk...} A_{1i}A_{2j}A_{3k}..., where epsilon is the n-dimensional Levi-Civita symbol (i.e it is equal to +1 if the indices are an even permutation of ijk... and -1 if the indices are an odd permutation of ijk..) and A_{ij} is an n by n matrix. In the expression for the determinant the rows (or columns) appear as products. Interchanging the order of these changes nothing but interchanging the order changes permutation of the indices of epsilon. Therefore, interchanging two rows you get a minus or a plus sign depending on if the interchange implies an odd or an even permutation of the indices of epsilon.