TASK DEFINITION: In this task you will be given a text passage from a trivia quiz. You should choose the category this question belongs to from these categories: History, Science, Social Science, Fine Arts, Literature. You can only choose one category, so if in doubt, please choose the closest match.
PROBLEM: Laplaces rule may be used to compute a version of this quantity for its namesake tensor in order to prove the Alon-Tarsi conjecture for Latin squares of even order; that proof uses Cayleys version of this operator. The Cauchy-Binet rule generalizes it to commutative rings, while Leibnizs formula uses permutations of elements to compute it. Setting to zero the result of this operation on lambda I minus A allows us to calculate As eigenvalues. Cramers rule uses ratios of these to solve systems of linear equations. For ten points, name this quantity that is a d minus c b for a 2 by 2 matrix a b c d.

SOLUTION: Science

PROBLEM: This entity shares its name with one controlled by the astronomer and investigator of mesmerism, Jean Sylvain Bailly. It saw the execution of General Lecomte, and Marshall MacMahon led a campaign against it that culminated with a mass execution at the Wall of the Federalists. Its aftermath saw a mass deportation to New Caledonia, and it was opposed by Adolphe Thiers. Louis-Auguste Blanqui was its nominal leader despite being imprisoned at the time, and it ended in the "Bloody Week" amidst fierce barricade fighting. Growing out of the Franco-Prussian War, for 10 points, name this socialist government that held power for two months in 1871 in the French capital.

SOLUTION: History

PROBLEM: Encouraged to become an artist by Sir Joshua Reynolds, he had a strong influence on his younger contemporary William Blake after his ten year foray to Italy to study the works of Michelangelo. Born in Zurich, he was appointed keeper of the Royal Academy in 1804, and is famous for his paintings and drawings of nude figures in strained poses showing intense emotion. FTP, name this Swiss-English painter, whose macabre fantasies included 1781's The Nightmare.

SOLUTION:
Fine Arts