(i) Let $X$ be an affine variety over an algebraically closed field. Define what it means for $X$ to be irreducible, and show that if $U$ is a non-empty open subset of an irreducible $X$, then $U$ is dense in $X$.

(ii) Show that $n \times n$ matrices with distinct eigenvalues form an affine variety, and are a Zariski open subvariety of affine space $\mathbb{A}^{n^{2}}$ over an algebraically closed field.

(iii) Let $\operatorname{char}_{A}(x)=\operatorname{det}(x I-A)$ be the characteristic polynomial of $A$. Show that the $n \times n$ matrices $A$ such that $\operatorname{char}_{A}(A)=0$ form a Zariski closed subvariety of $\mathbb{A}^{n^{2}}$. Hence conclude that this subvariety is all of $\mathbb{A}^{n^{2}}$.