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The tensor rank and border rank of the $3 \times 3$ determinant tensor isknown to be $5$ if characteristic is not two. In this paper, we show that thetensor rank remains $5$ for fields of characteristic two as well. We alsoinclude an analysis of $5 \times 5$ and $7 \times 7$ determinant and permanenttensors, as well as the symmetric $3 \times 3$ permanent and determinanttensors. We end with some remarks on binary tensors.

On the tensor rank of $3\times 3$ permanent and determinant