Science Fiction / Fantasy book involving a young woman hired to read Tarot cards for a local lord, ends with her killing his brother for him

I recalled the name of the world where it was set, Ivory, and that led me to The Gate of Ivory by Doris Egan.

THE SCIENCE OF MAGIC ---

is what lures people from all over the galaxy to the world of Ivory. Ivory, exotic and dangerous, where everything is for sale, and law exists only to protect the most powerful families. And to anthropology student Theodora of Pyrene, Ivory --- the one place in the known universe where magic works and where those who command it can control society --- is an irresistible world to explore on her all-too-brief vacation.

But cut off from her companions and her ship, attacked and robbed, Theodora's short pleasure trip soon becomes a terrifying odyssey. For on Ivory all her studies and training are useless, and Theodora is forced to turn to fortune telling to survive. To her amazement, she discovers that she actually is gifted with magical skill, a skill that will plunge her into deadly peril, when she becomes involved with Ran Cormallon, one of the most powerful sorcerers on Ivory --- and a man whose unknown enemies will stop at nothing, including spell-murdering Theodora, in their campaign to destroy Ran and seize his power!

I found I had an ebook copy on my computer and I got some of the ordering of things wrong. Theodora does her reading for Ran at the very beginning of the chapter, he's explicitly not part of the nobility (just from a very rich family), and the fight doesn't happen until later (although it does involve the "hotpencils").

I was laying down cards in the marketplace when I got the latest job offer. "Here comes money," Irsa, the vendor next to me had said and moved out of the way so as not to scare him off. So I'd given him his fortune, all the usual sort of nonsense, and out he came with this. I hadn't expected it of him; he looked too respectable. True, he hadn't mentioned the exact nature of this job. But I'd been in the Square long enough—I thought—to know what that meant.

"I might want to hire you," he repeated, as though he expected a dim-witted foreigner like me might need it said twice.

....

"Thank you, noble sir." I was as phony as any other market fortuneteller.

"I'm not a noble sir. 'Gracious' will do."

....

She shrugged. "It's a risk. If you—" A man in a red embroidered robe leapt suddenly on an older man in brown who had been fingering the bronze cups in an adjoining stall. The two fell, pushing Irsa's cart back and upsetting one of her piles of fruit. She pulled the cart back farther as they scrabbled on the ground. She was standing perfectly still, her eyes following two rolling pellfruit across the dirt. The man in red had a hotpencil, which he pressed against the other's temple. The victim's face contorted. "It's not fair," he said, in a shaking voice. "You weren't—supposed to—"

"I was within touching distance," said the man in red.

The other was dead. Irsa went to pick up her pellfruit, stooping with a look of disgust on her face. "Aristocrats, both of 'em," she said as she returned to me, one large round fruit in each hand. She raised an arm to wipe sweat off her face with the back of her robe. "You'd think they had better places for that sort of thing. Why, dear," she said to me, "you look a bit scretchy. It's all this sun, makes everything seem more important than it is. I know you weren't brought up to it, dear, but so what if two fools choose to end their quarrel in front of us? It's happened before, it will happen again. Ishin na' telleth!" She looked around for something to give me to cheer me up, and ended by giving me what she gave anyone in distress: a piece of fruit. It was all she had to offer.

Indeed, her predecessor died in a fire:

"I didn't get very far, though." When he didn't reply, I said, "Why did the cards become hot? And what about that building on fire—"

"A temporary problem, I'm sure—it may have something to do with your predecessor in this position. She was involved in an accident, which is why I had to find a replacement."

The martial art was named "The River"

The River. That was my first day, my introduction to the way of movement that was sa'ret. We did the Old River first, the flowing stretches and bends of a river that wound over well-known lands. And we did the Middle River, the balancer; you use the staff most in Middle River. We finished with the Young River, the current with wild leaps over rock and down canyons.

And indeed, she kills Eln, although it was an onyx cat that was used to cool hands, and I misremembered how it worked:

But Annurian of Kado Island knew very well. "Eln," I said, and I took my hand from the pack strapped to my side and tossed him a round, black object. He caught it neatly. It was the onyx cat. Fresh from my bare hands, the scent of it all on its stony skin: the tenderness and the understanding and the gratitude, the hatred and the wish to end all this. An emotional confusion more than equal to his own. He looked up from the cat, his dark eyes widened in surprise. "Theo, sweetheart—" he said. "I can't—" And he looked down at his chest, at the knife handle suspended there, and I looked down, too, at my empty fingers and the empty sheath under my robe. "Theo," he said again. The bluestone pendant was a fire against my skin. I ignored it.

....

Ran pulled me out of the arena. The Duelmaster was talking, making some kind of announcement. "The Cormallon matter has been settled privately," I remember him saying; "Thank you for your time and attention."

Search terms that got me there: novel ivory tarot