2
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Unfortunately in my problem $n <m$. Do you have any idea about that? $\endgroup$ – user160867 Nov 18 '15 at 12:44
  • $\begingroup$ If you can prove rank(A)<n or rank(B)<n, the same method still works.. $\endgroup$ – user219967 Nov 18 '15 at 12:51
0
$\begingroup$

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

$\endgroup$
  • $\begingroup$ As I said, I do not want to compute the determinant. It contains many terms in the summation. I just wanna know about the conditions on $A$ and $B$ such that $Det(AB)=0$. $\endgroup$ – user160867 Nov 18 '15 at 12:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.