RO-NMO 2017 Problem 112

Let be two natural numbers $n\ge 2, k,$ and $k n\times n$ symmetric real matrices $A_1,A_2,\ldots ,A_k.$ Then, the following relations are equivalent: $1) \left| \sum_{i=1}^k A_i^2 \right| =0$ $2) \left| \sum_{i=1}^k A_iB_i \right| =0, \forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right)$ $||$ denotes the determinant.