  We show that the border support rank of the tensor corresponding to
two-by-two matrix multiplication is seven over the complex numbers. We do this
by constructing two polynomials that vanish on all complex tensors with format
four-by-four-by-four and border rank at most six, but that do not vanish
simultaneously on any tensor with the same support as the two-by-two matrix
multiplication tensor. This extends the work of Hauenstein, Ikenmeyer, and
Landsberg. We also give two proofs that the support rank of the two-by-two
matrix multiplication tensor is seven over any field: one proof using a result
of De Groote saying that the decomposition of this tensor is unique up to
sandwiching, and another proof via the substitution method. These results
answer a question asked by Cohn and Umans. Studying the border support rank of
the matrix multiplication tensor is relevant for the design of matrix
multiplication algorithms, because upper bounds on the border support rank of
the matrix multiplication tensor lead to upper bounds on the computational
complexity of matrix multiplication, via a construction of Cohn and Umans.
Moreover, support rank has applications in quantum communication complexity.
