What is the Rank of Matrix $A$?

Question

Let $x$ and $y \in \mathbb{R}^{n}$ be non-zero column vectors, from the matrix $A=xy^{T}$, where $y^{T}$ is the transpose of $y$. Then the rank of $A$ is ?

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I am getting $1$, but need confirmation .

Answer 1

For every vector $v$, we have $Av = x y^Tv= x (y^Tv)$. $(y^Tv)$ is a scalar, so $Av$ is contained in the span of the one dimensional space generated by $x$.

Answer 2

Each row of $A$ is a multiple of $x$, so each pairs of rows in $A$ is linearly dependent. Hence, the rank of $A$ is at most 1. If $x=\mathbf{0}$ or $y=\mathbf{0}$, of course, the rank is $0$.