# Matrix products under which the determinant behaves multiplicatively

The determinant behaves multiplicatively with respect to the usual matrix product $$\det(AB) = \det(A)\det(B),$$ and also with respect to the Kronecker (or tensor) product of square matrices $$\det(A\otimes B) = \det(A)^q \det(B)^p,$$ when $A$ and $B$ are $p\times p$ and $q \times q$ matrices, respectively.

Are there other natural types of matrix products under which the determinant behaves multiplicatively? To be completely precise, the property I need is that the determinant of the product is $0$ if and only if the determinant of at least one of its factors is $0$.

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First of all, what do you mean by a "product"? Do you mean an algebra structure on $\oplus_{n=0}^{\infty}Mat(n,\Bbbk)$? –  Qfwfq Sep 19 '10 at 15:20
Thinking about the 'natural' part of the question, would this be a good formulation? Fix a group $G$ and $G$-modules $U$ and $V$ (over a ground field $K$, say). Classify the $G$-modules $W$ and bilinear $G$-module morphisms $\mu: End(U)\times End(V) \to End(W)$ with the property that $\det_W(\mu(A,B)) = c\ \det_U(A)^q\det_V(B)^p$ for nonzero $c$ and $p,q>0$? The special case of $G=GL(n,\mathbb{R})$ and $U=V=\mathbb{R}^n$, would encompass regular matrix multiplication, reversed matrix multiplication, and the tensor product, though there are other examples. –  Robert Bryant Jul 21 '11 at 19:44

Direct summation (taking a $p \times p$ matrix $A$ and a $q \times q$ matrix $B$ and returning a block-diagonal $(p+q) \times (p+q)$ matrix $A \oplus B := \begin{pmatrix} A & 0 \\\ 0 & B \end{pmatrix}$) also works:

$$\det(A \oplus B) = \det(A) \det(B).$$

One can debate whether this operation deserves to be called a "matrix product", though (for instance, it is not distributive over addition).

EDIT: Another (somewhat trivial) example is the reversed multiplication operation $(A, B) \mapsto BA$. More generally, if there was a linear automorphism $T$ on $Mat_n$ that preserved the singular variety $\{ A \in Mat_n: \det A = 0 \}$, one could conjugate the usual matrix multiplication operation by $T$. In the above example, $T$ is the transpose operation $T: A \mapsto A^t$. As another example, one could let $T$ be a left multiplication operator $A \mapsto SA$ for some invertible $S$, in which case the matrix multiplication operation becomes $(A, B) \mapsto ASB$, which also seems to work. One can combine the two and obtain another operation $(A, B) \mapsto BSA$. I'm not sure if these are the only examples that can be constructed by this method.

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The determinant of the product of two non square matrices is nicely expressed by the Binet-Cauchy formula: $$\det(AB) = \sum_I \det A_I \det B_I$$ Here $A$ is $n \times m$ and $B$ is $m \times n$ and the sum ranges over $n$-subsets $I$ of the numbers $\{1,2,...,m\}$. $A_I$ means "select columns of $A$ indexed by $I$" and $B_I$ means "select the rows of $B$ indexed by $I$". If either $A$ or $B$ has rank less than $n$ than the determinant of $AB$ is, thus, zero.