How many rank $1$ matrices in $\mathbb Z^{m\times n}$ are there if entries are restricted to $\{0,1\}$?
2026-02-23 03:29:32.1771817372
Number of rank 1 matrices with $0/1$ entries?
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Pick a nonzero vector $v \in \mathbb{R}^n$ consisting of only zeros and ones. The number of one/zero matrices for which row span is equal to $v$ is then $2^m-1$ because each row can be either $v$ or $0$ and it can't contain all zero vectors. Summing up over $2^n-1$ possible nonzero $v$ vectors we get the answer of: $$(2^n-1)(2^m-1)$$