In Sh'mot 7:20 the pasuk tells of how Moshe and Aharon did what Hashem commanded and raised the staff and hit the water (well, ultimately, Aharon did). But the pasuk reads " וַיָּ֤רֶם בַּמַּטֶּה֙ וַיַּ֤ךְ אֶת־הַמַּ֙יִם֙ אֲשֶׁ֣ר בַּיְאֹ֔ר" with the translation being "he lifted up the rod and struck the water in the Nile" (Sefaria) and "he raised the staff and struck the water that was in the Nile " (Chabad).
Each translation has bamateh meaning "the ____" but wouldn't that be "HAmateh" (as in 4:17)? Wouldn't the "ba" prefix indicate "with", "through" or "in" as in 7:17?
The Tur HaAroch cites the Ramban on the word bamateh as
Aaron raised the staff with his hand extending it in all the directions of the globe, exactly as commanded,
and the Haamek Davar focuses more on who performs the action
ומש״ה לא כתיב מי הרים כמש״כ בכ״מ אלא מדבר בה׳. שהוא הרים כ״י במטה משה
that Hashem raised the stick, as it were.
Are there other commentaries which focus on the grammatical use of the BA prefix here? I can imagine that it would mean that Moshe and Aharon are raised in stature through their action of listening to Hashem and raising the staff but that's just my personal take.
Is there any other explanation for the construction?