Consider a $2\times 3$ grid where each entry is one of $0$, $1$, and $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below: $$\begin{bmatrix}  1& 2 & 0\\ 2& 1 &0 \end{bmatrix}$$