Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
The Incarnate Word died. The God/Man died. Physically His humanity died. His divinity is un-separable from His humanity. Therefore God died. His divinity was in the grave with His flesh. His divinity was in Hades with His soul. You cannot separate His divinity from His humanity. He is of one essence with the Father. He is of one essence with us. St John Cassian said, "in the one Person of the Lord Jesus Christ we can speak of man as eternal, and of God as dead."
To put it specifically, the Ethiopian Orthodox Tewahedo Church fully affirms that by the fact of the Incarnation and the Union, the Divinity of Jesus Christfully participated
with the Death on the Cross. In our Liturgy of Saint Dioscoros we chant routinely, "The Immortal Died, died to quicken Death."
But Death should be understand as a process, as a place, and not necessarily a state of existing (or not-existing), and then in this context Jesus Christ, God-Man, in His full capacities as the Word, went into
Hell and rose the captives there. This is fundamental to our Tewahedo soteriology, that the very Divinity of Christ's Person is what raised Him from Death, and we have had our own internal Adoptionist heresies which is why we explain this in detail, that Jesus Christ was raised up by the power of His Divinity, not as a dead human being who was deified by Unction of Adoption of the Holy Spirit or the Father as others have taught. Frankly if folks explain that the Divine did not participate in the Death, we believe that they are suggesting a momentary
separation between the Divinity and Humanity of the unified Person of Jesus Christ, precisely what we in the Tewahedo Church have always condemned the Chalcedon doctrines of doing (IE, creating a separation or distinction between the Natures, hence why we only speak of One Incarnate Nature of Christ and never two in avoiding this kind of idea of separating divinity and humanity at the moment of Christ's death).
This is a Mystery, not meant to be fully understood, but rather experienced from the depth of our hearts. Could someone please explain the EO teaching in regards to the Descent of Christ into Hell to raise up the Saints if the Unionized Person of Christ, Divine and Human, did not go into Death?