b[matrix_] := Block[{r, k, choice, listofdet}, {r, k} = Dimensions[matrix]; choice = Subsets[Table[i, {i, r + k}], {r}]; listofdet = Table[ Det[ Join[ IdentityMatrix[r], Transpose[matrix] ] [[ choice[[i]] ]] ], {i, Binomial[r + k, k]}]; Count[listofdet, u_ /; u != 0]] (*r=3 and 11