Determinant Question: Understanding Row Interchange & Cycles

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In summary, the determinant of a matrix changes when the rows are interchanged, but remains the same when the rows are cyclically changed in order. This is due to the rule that the determinant is equal to the product of the matrix elements multiplied by the Levi-Civita symbol, which changes sign when the order of the rows is interchanged. This is a fundamental property of matrices.
  • #1
Weather Freak
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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!
 
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  • #2
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.
 
  • #3
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
 
  • #4
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.
 

1. What is a determinant?

A determinant is a mathematical concept that is used to determine various properties of a matrix. It is a numerical value that can be calculated from the elements of a square matrix.

2. How is the determinant of a matrix calculated?

The determinant of a matrix can be calculated in several ways, including using the traditional method of finding the product of the elements on the main diagonal and subtracting the product of the elements on the opposite diagonal. Other methods include using the cofactor expansion, Gaussian elimination, or using properties such as row interchange and cycles.

3. What is row interchange in terms of the determinant?

Row interchange, also known as row swapping, is a method used to manipulate the elements of a matrix in order to simplify the calculation of its determinant. It involves swapping the positions of two rows in a matrix, while keeping the elements in each row in the same order.

4. How does row interchange affect the value of a determinant?

Row interchange does not change the value of a determinant, but it can change the sign of the determinant. This means that if two rows are swapped in a matrix, the resulting determinant will be equal to the original determinant multiplied by -1.

5. What are cycles in terms of the determinant?

Cycles are a type of permutation that is used to manipulate the elements of a matrix in order to simplify the calculation of its determinant. A cycle involves moving the elements of a matrix in a circular pattern, with each element moving to the position of the next element in the cycle.

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