IMC 1995 · Problem 1

Let $A$ be $3 \times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for each column vector $u \in \mathbb{R}^3$. Prove that:

a) $A^\top = -A$, where $A^\top$ denotes the transpose of the matrix $A$;

b) there exists a vector $v \in \mathbb{R}^3$ such that $Au = v \times u$ for every $u \in \mathbb{R}^3$, where $v \times u$ denotes the vector product in $\mathbb{R}^3$.