Question 47
Let $A_1, A_2, \dots, A_k \in \mathcal{M}_n(\mathbb{R})$ be symmetric matrices. Prove that the following are equivalent:
1)
$$
\det(A_1^2 + A_2^2 + \dots + A_k^2) = 0;
$$
2)
for any matrices $B_1, B_2, \dots, B_k \in \mathcal{M}_n(\mathbb{R})$ we have
$$
\det(A_1 B_1 + A_2 B_2 + \dots + A_k B_k) = 0.
$$
1)
$$
\det(A_1^2 + A_2^2 + \dots + A_k^2) = 0;
$$
2)
for any matrices $B_1, B_2, \dots, B_k \in \mathcal{M}_n(\mathbb{R})$ we have
$$
\det(A_1 B_1 + A_2 B_2 + \dots + A_k B_k) = 0.
$$