Cauchy–Binet Formula

The Cauchy–Binet formula expands the determinant of a rectangular matrix product into a sum of corresponding minor products.

Interactive Experiment

Explore the relationship between the rectangular factors and their minors in the Cauchy-Binet Formula module:

1. Observe the initial state with the matrices from the worked example, noting that the sum of the minor products equals 11.
2. Modify the entries of A (e.g., set A12 to -1). Notice how the minor determinants can become negative, causing partial cancellation in the sum.
3. Try creating a rank-one matrix for A by setting the second row to be a multiple of the first row (e.g., set A21=2, A22=4, A23=0). Observe that all 2×2 minors of A become zero, and thus det(AB) collapses to 0.
4. Watch the dynamic updates in the "Cauchy-Binet Expansion" panel to see exactly which minor products contribute to the final determinant.