Etymology
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mother (n.1)

"female parent, a woman in relation to her child," Middle English moder, from Old English modor, from Proto-Germanic *mōdēr (source also of Old Saxon modar, Old Frisian moder, Old Norse moðir, Danish moder, Dutch moeder, Old High German muoter, German Mutter), from PIE *mater- "mother" (source also of Latin māter, Old Irish mathir, Lithuanian motė, Sanskrit matar-, Greek mētēr, Old Church Slavonic mati), "[b]ased ultimately on the baby-talk form *mā- (2); with the kinship term suffix *-ter-" [Watkins]. Spelling with -th- dates from early 16c., though that pronunciation is probably older (see father (n.)).

Sense of "that which has given birth to anything" is from late Old English; as a familiar term of address to an elderly woman, especially of the lower class, by c. 1200.

Mother Nature as a personification is attested from c. 1600; mother earth as an expression of the earth as the giver of life is from 1580s. Mother tongue "one's native language" is attested from late 14c. Mother country "a country in relation to its colonies" is from 1580s. Mother-love "such affection as is shown by a mother" is by 1854. Mother-wit "native wit, common sense" is from mid-15c.

Mother of all ________ (1991), is Gulf War slang, from Saddam Hussein's use in reference to the coming battle; it is an Arabic idiom (as well as an English one), for instance Ayesha, second wife of Muhammad, is known as Mother of Believers; the figure is attested in English in 19c. (Virginia is called mother of commonwealths from 1849). Mother Carey's chickens is late 18c. sailors' nickname for storm petrels, or for snowflakes.

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matrix (n.)

late 14c., matris, matrice, "uterus, womb," from Old French matrice "womb, uterus" and directly from Latin mātrix (genitive mātricis) "pregnant animal," in Late Latin "womb," also "source, origin," from māter (genitive mātris) "mother" (see mother (n.1)).

The many figurative and technical senses are from the notion of "that which encloses or gives origin to" something. The general sense of "place or medium where something is developed" is recorded by 1550s; meaning "mould in which something is cast or shaped" is by 1620s; sense of "embedding or enclosing mass" is by 1640s.

The mathematical sense of "a rectangular array of quantities (usually square)" is because it is considered as a set of components into which quantities can be set. The logical sense of "array of possible combinations of truth-values" is attested by 1914. As a verb, in television broadcasting, from 1951.

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matter (n.)

c. 1200, materie, "the subject of a mental act or a course of thought, speech, or expression," from Anglo-French matere, Old French matere "subject, theme, topic; substance, content; character, education" (12c., Modern French matière) and directly from Latin materia "substance from which something is made," also "hard inner wood of a tree." According to de Vaan and Watkins, this is from mater "origin, source, mother" (see mother (n.1)). The sense developed and expanded in Latin in philosophy by influence of Greek hylē (see hylo-) "wood, firewood," in a general sense "material," used by Aristotle for "matter" in the philosophical sense. 

The Latin word also is the source of Spanish, Portuguese, and Italian materia, Dutch, German, and Danish materie, vernacular Spanish madera, Portuguese madeira "wood" (compare Madeira). The Middle English word also sometimes was used specifically as "piece of wood."

From c. 1200 as "a subject of a literary work, content of what is written, main theme;" sense of "narrative, tale, story" is from c. 1300. Meaning "physical substance generally" is from mid-14c.; that of "substance of which some specific object is or may be composed" is attested from late 14c. Meaning "piece of business, affair, activity, situation; subject of debate or controversy, question under discussion" is from late 14c. In law, "something which is to be tried or proved," 1530s.

Matter of course "something expected" attested from 1739 (adjectival phrase matter-of-course "proceeding as a natural consequence" is by 1840). For that matter "as far as that goes, as far as that is concerned" is attested from 1670s. What is the matter "what concerns (someone), what is the cause of the difficulty" is attested from mid-15c., from matter in the sense of "circumstance or condition as affecting persons and things." To make no matter to "be no difference to" also is mid-15c., with matter in the meaning "importance, consequence."

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