Well, you also subtracted 1/2 of the first line from the third line!

If you "add 1/2 the first row to the third" you getHere's my confusion. To eliminate the first 1.5 I can either add 1/2 the first row to the third or -5.75/1.5 the second row to the third. Depending on which route I take the last element in the third row will be differnt, and when I multiply the numbers along the diagonal I will get 60 or 72 for the determinant.

Which is NOT triangular because you have added (1/2)(4)= 2 to the 0 that was in the first column of the third line.

That is why you should always work with the line that has only "0"s to the left of the pivot.