Does the fact that we are justified by faith alone imply that obedience to the law is no longer necessary?

No, for in Romans 3:31 Paul raises the question of whether the Torah is abrogated by means of faith. Do we then overthrow the law by this faith? and the answer is negative: By no means! The Torah is not the same as the law of works referenced in Romans 3:28. For one it contains promises about the coming Redeemer (Romans 1:2). The law and the prophets testify of Jesus Christ (Romans 3:21). For this reason Paul can say, On the contrary, we uphold the law. Because this statement at the end of this chapter is so brief it at first seems to be a rather bold assertion, but it is thoroughly backed up in Romans 4:1–25, which shows that this faith dates back to Abraham (even before his circumcision, Romans 4:1–12) and did not only begin with the Mosaic law which was given later (Romans 4:13–16). In this regard Paul says that the law brings wrath, for where there is no law there is no transgression (Romans 4:15). The law records and reveals the transgression. That is why it is a law, after all. Since trespasses are revealed through the law (see Romans 3:9–19), the law by itself, that is, isolated from the promise and from faith, would be reduced to a means of justifying God’s wrath (given our defective state). But since the law as it is has been given along with the promise, it is very much profitable. It effectively proves the value of Abraham already having received a promise and being made righteous through faith. Faith is not an emergency exit from the law; rather, the law confirms to us the safety of faith!¹