Suppose we have some arbitrary square matrices $ A,B $ and $C$ of dimension n. We can find the determinant of their sum, i.e. $$ y = \det(A+B+C) $$ My question is, how does the matrix $A$ effect the value of $y$? In other words, is it possible to know how much the value of $y$ will change w.r.t. to addition or removal of $A$.
My approach: I added $\phi_i$ (it is either 0 or 1) to the equation and tried to differentiate it w.r.t. $\phi_i$.
$$ y = \det(\phi_A A + \phi_B B + \phi_C C) $$ $$ \dfrac{\partial y}{\partial \phi_A} = \text{tr}(\text{adj}(y)A) $$ But I am unable to proceed any further from this. I don't know whether this result is correct or am I going in the right direction. What should I do?
Note: Take n = 2 or 3. Keep it small.