Determinant of a matrix. Why is it called an expansion of cofactors? Why is the word expansion used? What does it mean here?

 A: Frequently it happens that in a set of objects that can be composed together in some way, there are objects that are for some reason simpler than others.
In this circumstances it can be convenient to be able to find how simpler objects combine to give an object.
Once found, one can say that the object is decomposed over the simpler objects, but no insight into how simpler objects compose is given.
If the composition is a sum of the simpler objects each multiplied by a factor then one can use the espression "the object is expanded by the simpler objects".
(See e.g. vector expansion, Taylor or Fourier expansion)
If the composition is a product of the simpler objects each raised to some power then one can use the espression "the object is factored into the simpler objects".
(See e.g. integer factorization or polynomial factorization)

Cofactor = complement of factor
$C_{11}$ is the complement of the factor $a_{11}$. It is the complement because it is computed by complementation of $a_{11}$ w.r.t. the matrix, that is, by eliminating by the matrix the row and the column containing $a_{11}$.
