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Answer: There is no individual mitzva to eat five k’zeitim of matza. Rather, the fact that matza "wears different hats" makes it necessary to eat multiple k’zeitim – between three and five to be "less exact."
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Another machloket Rishonim (see Tosafot, Berachot 39b) is whether the same matza can be used for the two elements. One approach is that if one uses one matza despite the two berachot (and elements), it violates the rule to not do "mitzvot in bunches." A second approach is that it is enough to eat from one matza because there is one classic mitzva (eating matza). The beracha of Hamotzi is just a regular beracha on food, not a separate act of mitzva or a sign of one. The Shulchan Aruch (OC 475:1) rules that we must eat from separate matzot and adds that one requires a k’zayit from each. This is the "second k’zayit."
Some question why we would need a full k’zayit from the "matza of Hamotzi," considering that we make Hamotzi on any amount of bread. The Pri Chadash (ad loc., cited by the Mishna Berura 475:9) says it is because of a machloket whether the whole matza is for Hamotzi and the broken one is for the mitzva of matza, or vice versa. This causes us to treat each matza as if it is the one for the mitzva of matza, so that we need a k’zayit of each. Some say that this chumra is anachronistic. Since nowadays participants only receive a small amount from the "whole" and "broken" matzot held by the leader of the Seder, there is little purpose for more than a second full k’zayit (see Dirshu 474:8; Halichot Shlomo, Moadim I:9:40). Others try to uphold the minhag of two k’zeitim on various grounds (see Piskei Teshuvot 475:5). In any case, all major requirements are fulfilled by eating one k’zayit from any matza (Mishna Berura 575:11).
The next k’zayit is for korech, which is a remembrance of the way Hillel instructed people to eat matza and maror together when having a Seder with a Korban Pesach. For this, all agree that one k’zayit suffices.
The final eating is the afikoman. The Shulchan Aruch (OC 477:1) suffices with one k’zayit, but Acharonim bring a minhag to have two k’zeitim. Some Rishonim (see Rashi, Pesachim 119b) say afikoman is intended to be the main fulfillment of eating matza, but that does not explain two k’zeitim since there is no new beracha of Hamotzi on it. The Magen Avraham (477:1) says that since the afikoman is a remembrance of the Korban Pesach, (Rosh, Pesachim 10:34), we have one k’zayit to represent the korban and one representing the matza eaten with it. The Taz (477:1) says that we eat an especially big amount because the mitzva is dear to us. Either way, this minhag is a chumra, which is no more than preferable (Mishna Berura 477:1; Yalkut Yosef, Tzafun 1).
The stakes between the different opinions are small. Poskim agree that for the "second k’zayit" each time, one can follow a lenient opinion on the size of a k’zayit, and most of us will eat a much bigger shiur if we assume only k’zayit (see Ohr L’tziyon III:15:12). So our big k’zayit will come at least close to covering a basic two k’zeitim.
Rabbis in Midrash
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Kuzari -Rabbi Ari Shvat Kuzari class 9 - "Seeing is Believing" (parag. 21-30)
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