IMC 2004 · Problem 1

Let $A$ be a real $4 \times 2$ matrix and $B$ be a real $2 \times 4$ matrix such that

$$AB = \begin{pmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{pmatrix}.$$

Find $BA$.