Assume $\int^a_b \operatorname{tr}(A(t)B(t))~dt=0$ for any $B$, where $A,B$ are $n \times n$ matrices. Does this imply $A=0$?

Question

Assume $\displaystyle\int^a_b \operatorname{tr}(A(t)B(t))~dt=0$ for any $B$, where $A,B$ are $n \times n$ matrices.

Does this imply $A=0$?

If this is not true, can we add some conditions for $A, B$ to make the proposition true? For example, add some conditions like $A$ is symmetric, $A$ is skew-symmetric, both $A~\text{and}~B \in SO(n)$, etc.

Answer 1

Following Math Lover's hint, consider $B(t)$ being $A(t)^\top$. Then check that the integrand $\text{tr}(A(t) A(t)^\top)=\sum_i \sum_j A(t)_{ij}^2$ is nonnegative.