I would like to know the difference between cofactor expansion and Laplace expansion. It looks like they are the same thing under different names. However, I was told that Laplace expansion is more general than cofactor expansion. I could not get an elaboration on it.
Is it correct that they are different things? If so, can anyone tell me the difference?
The expansion along one row or column ts known by various names, including "co-factor expansion" and "Laplace expansion." You are probably thinking of the "generalized Laplace expansion" along several rows or columns at once. Specifically, taking the row case and expanding along rows $i_1,i_2,...,i_m$ of the $n \times n$ matrix $A$ over a commutative rtng, the determinant of $A$ is the sum of all products obtained by taking the determinant of the matrix formed from rows $i_1,i_2,...,i_m$ and columns $j_1,j_2,...,j_m$ of $A$ and multiplying by the complementary co-factor, i.e. $$(-1)^{i_1+...+i_m+j_1+...+j_m}\det(A \text{ with rows}i_1,...,i_m\text{ and columns }j_1,...,j_m \text{ removed.})$$