Formal:
* The judge delivered an authoritative ruling on the case, leaving no room for doubt.
* Her research on climate change is highly regarded and considered authoritative in the scientific community.
* The textbook presented an authoritative overview of the historical events leading up to the revolution.
Informal:
* My grandfather has an authoritative way of telling stories that always keeps us captivated.
* The new CEO has a very authoritative presence, which inspires confidence in the employees.
* The doctor's tone was authoritative as he explained the treatment plan, assuring me that everything would be alright.
Figurative:
* The old oak tree stood authoritative over the clearing, its branches reaching towards the sky like the arms of a wise old man.
You can also use "authoritative" to describe something that is respected or influential in its field:
* The New York Times is an authoritative source of news.
* The book is considered an authoritative guide to starting a business.
The key thing to remember is that "authoritative" implies commanding respect and credibility, whether it's a person, a source of information, or even an object.