Tearing is mesaken (Shabbos 105b)

The gemara quotes a braisa that says that if one tears his garment on Shabbos because a relative died then he is yotze the chiyuv kria but is also chayav for kria on Shabbos. How is he yotze the mitzva if it's a mitzva habaa b'aveira? Why is this not different than eating stolen matza? Someone asked this question this morning and I saw that Kollel Iyun Hadaf asked the question and suggested a couple answers:

1. Some Rishonim explain that when one steals Matzah to eat on Pesach, the transgression has an effect on the object (the Matzah), and that effect remains on the Matzah until the moment he eats it. A stolen piece of Matzah is subject to an obligation to return the stolen object to its rightful owner, and therefore the object itself is an "object of Aveirah." In contrast, when a person tears his clothing upon hearing of the death of a relative on Shabbos, the transgression has no effect on the object, and thus one fulfills his obligation. The concept of "Mitzvah ha'Ba'ah b'Aveirah" applies only when the object is affected by the transgression and is an object whose very nature contains an element of transgression. (RITVA to Sukah 30a)

I saw that the Sfas Emes quotes this answer also. He also uses this to explain an apparent contradiction within the halachos of kria. The Tur paskens (Y"D 340:28) that you are not yotze kria if you do it on a stolen garment but you are if you tear a garment of yours on Shabbos. This makes sense based on this answer. If you tear on a stolen object then the object itself is a stolen object so you can't be yotze the mitzva of kria with that but here it's just the act which is forbidden but the object never becomes a forbidden object.

2. Other Rishonim explain that "Mitzvah ha'Ba'ah b'Aveirah" applies only when the Aveirah itself is the cause of the fulfillment of the Mitzvah. That is, without the Aveirah, the Mitzvah would not be done (see TOSFOS to Sukah 30a). A person must own the Matzah he eats in order to fulfill the obligation to eat Matzah on Pesach night. When a person steals Matzah, he acquires it when the owner gives up hope of ever retrieving (Yi'ush). Thus, it is the Aveirah (the act of stealing) that causes the Matzah to become his and enables him to fulfill the Mitzvah. In contrast, in the case of one who tears his clothing on Shabbos in mourning, the transgression of tearing on Shabbos is not what enables him to fulfill the Mitzvah. Since the person could fulfill his obligation to tear this garment without transgressing Shabbos (such as by tearing it on Sunday), his act on Shabbos is not considered a "Mitzvah ha'Ba'ah b'Aveirah."

The Sfas Emes didn't like this answer so much because the split is hard to understand. Just like you could fulfill your obligation by tearing on Sunday so too you could fulfill by tearing another garment. Why is one considered a mitzva habaa b'aveira but not the other? He also suggests another couple of answers but doesn't stick with either one. He likes the first answer quoted above the best.