Correlation between rank and determinant

Question

What can we say about a matrix which has the following property: $\det(A)=\operatorname{rank}(A)$? My guess is: if matrix A doesn't have a full rank that means that $\det(A)=0$ so that is not really an interesting case. If it does have full rank, that means $\det(A)\ne 0$ so it does mean that matrix A is regular but is there any theorem or some hint which I can come to a more concrete, stronger conclusion from than this? And is there a non-zero matrix which has some additional properties except $\operatorname{rank}(A)=\det(A)$?