Pensieve header: Det on polys has a bottleneck in dimension 11. See also http://mathematica.stackexchange.com/questions/109631/issue-in-det-when-computing-determinants-of-polynomials.

n=12;
M=Table[RandomInteger[10]t+RandomInteger[10],{i,1,n},{j,1,n}]

Timing[Det[M]]

Timing[Table[
Det[Delete[j]/@Delete[i]@M],
{i,n},{j,n}
]] // Short

Table[Echo@{n->First@Timing@Table[Det[Table[RandomInteger[10]t+RandomInteger[10],{i,1,n},{j,1,n}]], {100}]},
{n,20}];