  Let $M$ be an $n\times n$ matrix of homogeneous linear forms over a field
$\Bbbk$. If the ideal $\mathcal{I}_{n-2}(M)$ generated by minors of size $n-1$
is Cohen-Macaulay, then the Gulliksen-Neg{\aa}rd complex is a free resolution
of $\mathcal{I}_{n-2}(M)$. It has recently been shown that by taking into
account the syzygy modules for $\mathcal{I}_{n-2}(M)$ which can be obtained
from this complex, one can derive a refined signature-based Gr\"obner basis
algorithm DetGB which avoids reductions to zero when computing a grevlex
Gr\"obner basis for $\mathcal{I}_{n-2}(M)$. In this paper, we establish sharp
complexity bounds on DetGB. To accomplish this, we prove several results on the
sizes of reduced grevlex Gr\"obner bases of reverse lexicographic ideals,
thanks to which we obtain two main complexity results which rely on conjectures
similar to that of Fr\"oberg. The first one states that, in the
zero-dimensional case, the size of the reduced grevlex Gr\"obner basis of
$\mathcal{I}_{n-2}(M)$ is bounded from below by $n^{6}$ asymptotically. The
second, also in the zero-dimensional case, states that the complexity of DetGB
is bounded from above by $n^{2\omega+3}$ asymptotically, where $2\le\omega\le
3$ is any complexity exponent for matrix multiplication over $\Bbbk$.
