The sign of a determinant changes when two rows are switched due to the underlying properties of linear transformations and the volume they represent.

Explanation:

1. Geometric Interpretation: Determinants represent the scaling factor of the transformation described by a matrix. This scaling factor can be interpreted as the signed volume (or area, in 2D) of the parallelepiped (or parallelogram) formed by the row vectors of the matrix.

   • If you swap two rows of the matrix, you effectively "flip" the orientation of the parallelepiped. This changes the sign of the determinant but not its magnitude, as the volume remains the same. Flipping corresponds to a change in orientation, which is represented by a change in sign.

2. Algebraic Interpretation: The determinant is a sum of products of matrix entries, with each term having a sign determined by the number of row/column swaps necessary to place the matrix into its canonical (identity) form.

   • Swapping two rows corresponds to multiplying by $-1$, because one pair of terms in each permutation will swap. Mathematically, each time two rows are exchanged, the permutation of the matrix's rows is an odd permutation, which changes the sign of the determinant.

In both interpretations, row swapping fundamentally alters the structure or orientation of the matrix's representation, leading to a sign change in the determinant.

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Would you like more details on this topic or have further questions?

Here are 5 related questions to expand the discussion:

1. How does the determinant relate to the eigenvalues of a matrix?
2. What happens to the determinant when multiplying a row by a scalar?
3. Can determinants be applied to non-square matrices?
4. How do column swaps affect the determinant, and why?
5. How does the cofactor expansion work for calculating determinants?

Tip: Always remember that the determinant of a triangular matrix is simply the product of its diagonal elements!