Expansion Minimization

The determinant of a square matrix $A$ is typically defined in terms of cofactor expansion along the first row of $A$.  Then cofactor expansion along other rows or along columns is defined, and shown (or stated) to be equal to the determinant.  Following that, there is usually an example of a matrix that contains at least one zero.  This example is used to illustrate that sometimes, if we choose the appropriate row or column to expand along (namely, one with zeros), we can reduce work needed to compute the determinant using cofactor expansion.

It is certainly easy to tell which row(s) or column(s) to expand along to minimize the number of cofactors it is necessary to compute at each step.  Simply expand along a row or column that has the most zeros.

But what row or column should we expand along to minimize the total number cofactors to compute overall?  Or, to put it in more practical terms, what row or column should we expand along to minimize the
 amount of writing necessary?

Something to consider is that there are two ways to get a zero in the cofactor expansion.  If $B$ is some square submatrix encountered in the cofactor expansion of $A$, then either $b_{ij}=0$ or $\det B_{ij}=0$ ($B_{ij}$ being the $(i,j)-minor of $B$) to make a cofactor.  Since determining whether or not $\det B_{ij}=0$ requires a determinant computation, we would probably want to know the answer to the question to answer the question, which leads to circular reasoning.  So we'll assume that we're basing our expansion minimization strategy purely on which entries of $A$ are 0.

Of course, the answer to the question posed here is mostly useless for practical purposes, since nobody is going to compute the determinant through cofactor expansion for all but the smallest matrices.


But since we frequently present these example and pose these problems to students, with the expectation that they'll know what the best row or column is to expand along, I wonder if there's actually some way to tell what the best path to the determinant is from the matrix itself.