You should wrap your data into a list

{{6., 7.}, {6., 4.}, {1., 6.}, {4., 6.}, {7., 5.}, {6., 2.}, {6., 7.}, {6., 4.}}

and then apply the replacement rule outside the final bracket:

  {{6., 7.}, {6., 4.}, {1., 6.}, {4., 6.}, {7., 5.}, {6., 2.}, {6., 7.}, {6., 4.}}/. {a_, b_} :> {a, b, If[a >= b, 1, 0]}

The repacement rule instructs Mathematica to replace any list of two elements of the form {a,b} with the triplet {a,b,variable}, where the variable is 1 or 0 according to the values of a,b. The underscores in {a_, b_} mean that a and b are not meant to be literally the two symbols a and b, but they are patterns to which we assign the tags a and b for the purposes of the replacement.

The rule {a_, b_} :> {a, b, If[a >= b, 1, 0]} fails if your matrix is 2 by 2, in which case you should use the more elaborate version {a_?NumericQ, b_} :> {a, b, If[a >= b, 1, 0]} ,which checks if a is a number.

There are other ways of solving your problem, that may be more obscure but a lot faster for very large matrices.

What is the format of your matrix? Is it a list of lists? Can you give a small sample?

Something like this?

{{1, 2}, {2, 1}} /. {a_?NumericQ, b_} :> {a, b, If[a >= b, 1, 0]}