Exercise 8, § 10

Let $X = (\alpha_{ij})$ be a matrix with $m$ rows and $n$ columns over a field $K$. For $X$ to be of rank 1, it is necessary and sufficient that there exist in $K$ a family $(\lambda_i)_{1 \leq i \leq m}$ of $m$ elements not all zero and a family $(\mu_j)_{1 \leq j \leq n}$ of $n$ elements not all zero such that $\alpha_{ij} = \lambda_i \mu_j$ for every ordered pair of indices.