Determinant of multiplication of two nonsquare matrices

Question

Suppose $A$ and $B$ are $n\times m$ and $m \times n$ matrices, respectively, where $n<m$. The determinant of the product of two rectangular matrices can be obtained by the "Cauchy–Binet formula".

I do not need to compute the determinant of $AB$. I would like to just know when $Det(AB)=0$?

Can anyone helpe me please?

Answer 1

When $n>m$, we have $rank(AB)\le min\{rank(A),rank(B)\}\Rightarrow rank(AB)\le m\Rightarrow Det(AB)=0$

Answer 2

Let A be an m×n matrix and B an n×m matrix. Write [n] for the set { 1, ..., n }, and $\tbinom{[n]}m$ for the set of m-combinations of [n] (i.e., subsets of size m; there are $\tbinom nm of them)$. The Cauchy–Binet formula then states

$\det(AB) = \sum_{S\in\tbinom{[n]}m} \det(A_{[m],S})\det(B_{S,[m]})$ .a good source would be Wikipedia