Exercise 111.55.5. Let $k$ be an algebraically closed field. Let

\[ V_0 = \{ A \in \text{Mat}(3 \times 3, k) \mid \text{rank}(A) = 1\} \subset \text{Mat}(3 \times 3, k) = k^9. \]

Show that $V_0$ is the set of closed points of a (Zariski) locally closed subset $V \subset \mathbf{A}^9_ k$.

Is $V$ irreducible?

What is $\dim (V)$?

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