# Which $m \times n$ matrix is the product of an $m \times 0$ and $0 \times n$ matrix? [closed]

If we multiply a $$m \times 0$$ matrix with a $$0 \times n$$ matrix, which $$m \times n$$ matrix is the result? Is it the $$m \times n$$ matrix with all zeroes? Or is it some other matrix?

• This does not really make sense since m x 0 and 0 x n are empty matrices. Jul 4, 2023 at 15:11
• The obvious sometimes escapes one. I would go with zeroes. Jul 4, 2023 at 15:14
• @jjagmath I saw such "empty"-definitions so often (some cases were far stranger than the cases here) that I thought mathemticians would define it this way. Would be good , if this is not the case here , since everything can be exaggerated. Jul 4, 2023 at 15:27
• If this answer is valid, then for a given field $\mathbb{F}$: (1) for all $m$ there is exactly one $m \times 0$ matrix, which can be viewed as a sequence of $m$ 'rows' of elements of $\mathbb{F},$ each 'row' being a sequence of length $0;$ (2) there is exactly one $0 \times n$ matrix, which can be viewed as a sequence of $n$ 'columns' of elements of $\mathbb{F},$ each 'column' being a sequence of length $0;$ (3) therefore each element of the product is the sum of a sequence of length $0$ in $\mathbb{F},$ which by convention is $0.$ Jul 4, 2023 at 15:53
• As noted in this other answer, an $m \times 0$ matrix represents a linear transformation from a $0$-dimensional vector space to an $m$-dimensional vector space. There is only one such linear transformation: the zero map. Matrix multiplication corresponds to composition, and composing the zero linear transformation with any linear transformation will result again in zero. So yes, it's the zero matrix. Jul 4, 2023 at 18:46

Well, by all means it's the zero matrix. Specifically, $$[AB]_{h,k}=\sum_{j=1}^0 A_{h,j}B_{j,k}=0$$
since $$\sum_{j=1}^0$$ is an empty sum.
This is also consistent with $$A$$ representing the zero map $$\Bbb F^0\to \Bbb F^m$$ and $$B$$ representing the zero map $$\Bbb F^n\to\Bbb F^0$$.
In order to make sense of matrices with size zero, you need to treat $$m\times n$$ matrices as a triple $$(A,m,n)$$, where $$A$$ is an function in $$\Bbb F^{\{1,\cdots,m\}\times\{1,\cdots,n\}}$$ and $$m,n$$ are the two aforementioned natural numbers. The piece of information conveyed by $$m$$ and $$n$$ is usually unnecessary, since for $$m,n$$ positive the function $$(m,n)\mapsto\{1,\cdots,m\}\times\{1,\cdots,n\}$$ is injective, but, if you are to allow $$m=0\lor n=0$$, then you need to distinguish when the empty matrix is the representation in coordinates of a map $$\Bbb F^n\to \Bbb F^0$$, $$\Bbb F^0\to \Bbb F^m$$ or $$\Bbb F^0\to\Bbb F^0$$.
For the determinant, the zero matrix $$0\in\Bbb F^{0\times 0}$$ is treated as the identity of $$\Bbb F^0$$, therefore its determinant is $$1$$.