completely polarized polynomial

Question

Let $A$ be a $(r \times r)$-matrix. From the equation$$ \det\left(1+A\right)=\sum_{0\leq j \leq r} {r \choose j} H_j (A) $$ where $H_j (A)$ are homogenous polynomials of order $j$ in the entries of the matrix $A$. We define the the polarization $P_l$ of $H_l$,to be a $l$-multilinear, symmetric polynomial, satisfying $P_l(A, \cdots , A) =H_l (A)$, is there a general formula to write $P_l(A_1, ... , A_l)$ in terms of the $H_l$?

Answer 1

I think I have the answer: $$ P_l(A_1,\cdots , A_l)=\frac{(-1)^l}{l!}\sum_{j=1}^l \sum_{i_j < \cdots < i_j} (-1)^j H_l (A_{i_1} + \cdots + A_{i_j}). $$