en_tn/1ki/13/34.md

This matter became sin to the family of Jeroboam

This can be stated in active form. Alternate translation: "Jeroboam's family sinned by doing this thing" (See: rc://en/ta/man/translate/figs-activepassive)

This matter

This phrase refers to Jeroboam's setting up shrines and appointing priests.

caused his family to be destroyed and to be exterminated

This can be stated in active form. Alternate translation: "because of this God destroyed and exterminated Jeroboam's family" (See: rc://en/ta/man/translate/figs-activepassive)

to be destroyed and to be exterminated

These mean nearly the same thing. Alternate translation: "to be completely destroyed" (See: rc://en/ta/man/translate/figs-doublet)