  We prove that the determinantal complexity of a hypersurface of degree $d >
2$ is bounded below by one more than the codimension of the singular locus,
provided that this codimension is at least $5$. As a result, we obtain that the
determinantal complexity of the $3 \times 3$ permanent is $7$. We also prove
that for $n> 3$, there is no nonsingular hypersurface in $\mathbf{P}^n$ of
degree $d$ that has an expression as a determinant of a $d \times d$ matrix of
linear forms while on the other hand for $n \le 3$, a general determinantal
expression is nonsingular. Finally, we answer a question of Ressayre by showing
that the determinantal complexity of the unique (singular) cubic surface
containing a single line is $5$.
