Putnam 1959 A6

Let $m$ and $n$ be integers greater than $1$ and $a_1 ,a_2 ,\ldots, a_{m+1}$ be real numbers. Prove that there exist real $n\times n$ matrices $A_1 ,A_2,\ldots, A_m$ such that (i) $\det(A_j) =a_j$ for $j=1,2,\ldots,m$ and (ii) $\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.$