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We show that the r-by-r minors of a d-by-n matrix of variables form atropical basis of the ideal they generate if and only if r<4, or r=min(d,n), orelse r=4 and min(d,n)<7. This answers a question asked by M. Chan, A. Jensen,and E. Rubei.

When do the r-by-r minors of a matrix form a tropical basis?