In 1925 and 1926 two versions of quantum mechanics were born almost at once. Werner Heisenberg’s used infinite matrices; Erwin Schrödinger’s, a wave equation. Both gave the same results in experiments, but they seemed to speak of different things, and physicists used tricks that mathematicians found unacceptable, such as Dirac’s delta “function”. In Göttingen, working with Hilbert, von Neumann showed that the two were the same theory written in different coordinates, and rewrote everything in a single language. The book Mathematical Foundations of Quantum Mechanics, of 1932, is still the reference for anyone who wants the theory with full rigour.
Determinants were born to solve systems of linear equations. In 1750 Gabriel Cramer published the rule that bears his name, which gives each unknown as a ratio of two expressions formed from the coefficients. But an organized way of computing those expressions when there were many unknowns was missing. In 1772, almost at the same time, Alexandre-Théophile Vandermonde and Laplace studied these expressions as objects in their own right. The word determinant, in its present sense, was fixed only by Cauchy in 1812, and the vertical bars we use today came from Arthur Cayley in 1841.
Every quaternion $q = a + v$, with scalar part $a$ and vector part $v$, satisfies a quadratic equation with real coefficients. Since $v^2 = -\lvert v \rvert^2$ (see vectors), $$q^2 = a^2 + 2av + v^2 = a^2 - \lvert v \rvert^2 + 2av,$$and so $$q^2 - 2a\,q + \lvert q \rvert^2 = 0, \qquad \lvert q \rvert^2 = a^2 + \lvert v \rvert^2.$$A quaternion is a root of a real polynomial of degree 2, just as a complex number $z$ is a root of $z^2 - 2\operatorname{Re}(z)\,z + \lvert z \rvert^2 = 0$. In the Lectures on Quaternions of 1853 Hamilton went further and showed that certain linear functions of quaternions also satisfy a polynomial equation of their own, which he used to compute their inverses.