Adam Clarke on Matthew 16:16

Thou art the Christ, the Son of the living God - Every word here is emphatic - a most concise, and yet comprehensive, confession of faith.

The Christ, or Messiah, points out his divinity, and shows his office; the Son - designates his person: on this account it is that both are joined together so frequently in the new covenant. Of the living God Του Θεου, του ζωντος, literally, of God the Living One. The C. Bezae has for Του ζωντος the Living One, Του σωζοντος, the Savior, and the Cant. Dei Salvatoris, of God the Savior.

Living - a character applied to the Supreme Being, not only to distinguish him from the dead idols of paganism, but also to point him out as the source of life, present, spiritual, and eternal. Probably there is an allusion here to the great name יהוה Yeve, or Yehovah, which properly signifies being or existence.