HMMT 2018 C1

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.

$$\left[\begin{array}{lll} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array}\right]$$