A student in a linear algebra class looks at her homework problem. It says: Prove that the product $BA$ is the same for any $3\times 2$ matrix $A$ and $2\times 3$ matrix $B$ such that

$$AB=\begin{bmatrix} 3 &-1 & 0\\3 &-1 & 0\\-1 & 1 & \end{bmatrix}.$$

Unfortunately, the entry in the lower right corner of $AB$ is missing. Find the missing entry and solve the homework problem.