If I have a square invertible matrix and I take its determinant, and I find that $\det(A) \approx 0$, does this imply that the matrix is poorly conditioned?
Is the converse also true? Does an ill-conditioned matrix have a nearly zero determinant?
Here is something I tried in Octave:
a = rand(4,4); det(a) %0.008 cond(a)%125 a(:,4) = 1*a(:,1) + 2*a(:,2) = 0.000000001*ones(4,1); det(a)%1.8E-11 cond(a)%3.46E10