Three open dense subsets of the space of matrices

Method. If $p(A)$ is a polynomial in the $n^2$ entries of $A$ and $p(A_0)\ne0$ for one $A_0$, then $p\not\equiv0$ and $\{p\ne0\}$ is open and dense.

1. Invertible matrices. $p=\det$ is a polynomial in the entries and $\det I=1\ne0$. The set is $\{\det\ne0\}$.

2. $n$ distinct eigenvalues. The matrix has $n$ distinct complex eigenvalues iff the characteristic polynomial $\chi_A$ has $n$ distinct roots iff its discriminant $\operatorname{Disc}(\chi_A)\ne0$. The discriminant is a polynomial in the coefficients of $\chi_A$ (it is $\pm$ the resultant of $\chi_A$ and $\chi_A'$ divided by the leading coefficient $1$), and these coefficients are polynomials in the entries of $A$. So $p(A)=\operatorname{Disc}(\chi_A)$ is a polynomial in the entries. At $A_0=\operatorname{diag}(1,2,\dots,n)$ the eigenvalues are distinct, so $p(A_0)\ne0$.

3. All principal minors nonzero. A principal minor is the determinant of the submatrix obtained by keeping rows and columns indexed by a nonempty subset $S\subset\{1,\dots,n\}$. The product $p(A)$ of the $2^n-1$ principal minors is a polynomial in the entries, and $\{p\ne0\}$ is exactly the set where all of them are nonzero. For $A_0=I$, every principal minor equals $1$, so $p(I)=1\ne0$.

Conclusion. In each case the set is $\{p\ne0\}$ for a nonzero polynomial, hence open and dense.

Remark. "Generic" matrices are therefore invertible, diagonalisable over $\mathbb{C}$, and have nonzero principal minors: any claim that is closed and true on such matrices extends to all matrices by continuity. Lesson. Memorise the three standard dense open sets.