Lesson 6

In the proof, we will need the following theorem, showing how to find the determinant of a matrix that is the product of two rectangular matrices.

Theorem (Binet, Cauchy, 1812). Let \(A \in \mathbb{R}^{n \times m}\) and \(B \in \mathbb{R}^{m \times n}\), where \(m \ge n\). For \(S \subseteq [m]\), let \(A[S]\) (\(B[S]\)) denote the submatrix of \(A\) (respectively, \(B\)), corresponding to the columns (rows) \(S\). Then \[\begin{align*}\det(AB)=\sum_{S \in \binom{[m]}{n}}\det(A[S])\det(B[S]) \ .\end{align*}\]