Matthew 24:8 and
GRK: δὲ ταῦτα ἀρχὴ ὠδίνων
NAS: these things are [the] beginning of birth pangs
GRK: αὐτὸν τῇ ἀρχῇ καὶ τῇ
NAS: that they [could] deliver Him to the rule and the authority
GRK: ἐστιν [ἡ] ἀρχή πρωτότοκος ἐκ
NAS: the church; and He is the/[the] beginning, the firstborn
GRK: ἀληθινός ἡ ἀρχὴ τῆς κτίσεως
NAS: Witness, the Beginning of the creation
GRK: ἡ ἀρχὴ καὶ τὸ…
NAS: the beginning and the end.
If the article is present, arche might well be functioning qualitatively or again as “the ruler”. See how in Greek it immediately precedes the word for “firstborn”. In English we separate the two with a comma (“…he is the beginning, the firstborn from among the dead”), but this qualitative use could allow for the article to be attached to "the firstborn" rather than "the beginning". However, an even more plausible explanation can be found, in my eyes, when we integrate how arche is generally used in in this epistle. Bearing in mind the shrinking numbers of New Testament scholars still attributing the epistle to genuine Pauline authorship, we should not take lightly that ALL the other declinations of arche in Colossians refer to rule, rulers, principalities, and so on. Colossians 1:18 would be the only exception. If this case were to be worked out more fully, then a translation could legitimately go more along the lines of he is the ruler and the firstborn from among the dead. In either case, the inclusion of the article could thus be explained.
If the article is not present, then Colossians 1:18 would definitely read THE beginning and would be in line with the overall New Testament usage that we are observing.