The connection with linear systems
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The first study of a linear system—in this case, three equations in two unknowns—was carried out by Leibniz in the 1670s. In his Treatise of Algebra, published posthumously in 1748, Colin MacLaurin examines systems with up to four unknowns; quantities resembling modern determinants appear there. Gabriel Cramer subsequently defines the determinant as we know it, but calculating it is cumbersome because matrix notation (using arrays) did not yet exist. Étienne Bézout improves on this work and proves in 1764 that a system of n equations in n unknowns has nonzero solutions if and only if its determinant does not vanish. In 1772, Alexandre Vandermonde and Pierre-Simon Laplace devise methods for expanding determinants. The word determinant itself first appears in 1801, in Latin, in the work of Carl Friedrich Gauss. Louis-Augustin Cauchy was already writing determinants as arrays in 1815, but the modern notation between two vertical bars was Arthur Cayley's idea, introduced in 1841.
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A fruitful geometric interpretation
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Without a unit of measurement, we cannot define the length of a line segment. We can, however, calculate the ratio of the lengths of two line segments. Similarly, for two nonzero collinear vectors u and v in a vector space, there is a scalar λ such that v = λu; the scalar λ is the "algebraic ratio of the two vectors" (λ < 0 if the vectors point in opposite directions). The determinant generalizes this property to higher dimensions.
