Question: Why can one fulfill the mitzvah of eating matzah on the seder night let say if it fell out in Shabbos with matzah that he carried in a public domain that night without an eruv but he cannot fulfill the mitzvah with stolen matzah?
A. The Mishna Berura (454:4:14, 15) says a person doesn’t even technically fulfill the mitzvah of eating matzah and would have to eat again and say a new bracha if one uses stolen matzah. But if one stole wheat or flour and made matzah he can fulfill the mitzvah because he acquired it with a change in the substance and he just has to pay the owner for the ingredients he stole.
B. The first Beur Halacha in Mishna Berura siman 318 says that if you purposefully carry on Shabbos you can still use it after Shabbos though if you purposefully cooked on Shabbos you can never use it because the Rabbis fined the one who purposefully transgressed Shabbos in a way which will change the item used to do a sin.
Answer: The Be’ur Halacha “vi’li’inyan bracha etc.” Says that the Poskim write that specifically by stolen matza one does not fulfill the mitzvah because the matzah itself came through sin. (According to Rebbe Akiva Aiger putting it into the mouth to eat has to be the means of stealing the object while fulfilling the mitzvah at the same time.) On the other hand, a carrying matza from the private domain to the public domain when Pesach falls out on Shabbos, one can still fulfill his mitzvah with that matza and even say a blessing for it because he is the one that actively transgressed a mitzvah, but the matzah itself isn’t a sin. This means that the matzah itself is deemed stolen matzah hence it’s an object of sin. But the matza that is carried is not deemed to be a status of a sin, it just that the action you did is a sin, totally disconnected from the object therefore when fulfilling the mitzvah it not through a sin in terms of carrying on Shabbos.