= Solution
{c}
<Alexandre-Théophile Vandermonde> presents a programme in <elimination theory>: remove <unknowns> from simultaneous <polynomial equations> in a way that works uniformly as the number of <equations> and their <polynomial degrees> vary. The immediate achievement is narrower than that programme: a general construction for <linear equations>. The academy year 1772 used in the exam is not the printing date of the memoir, which appeared in 1776. Its compact indices distinguished the equation containing a <coefficient> from the <unknown> it multiplied; they were not powers. That notational separation is crucial to interpreting the work.<Mémoire sur l'élimination by Alexandre-Théophile Vandermonde>[Opening, printed pp. 516–517; academy year 1772, printed 1776]{ref}<A case of mathematical eponymy: the Vandermonde determinant by Bernard Ycart>[§2.1, discussion and reproduction of Vandermonde's coefficient notation; bibliography]{ref}
In modern language take a <system of linear equations> $A\mathbf{x}=\mathbf{b}$ with $n$ <unknowns>. Its <determinant> is an alternating expression in the <coefficients>:
$$
D=\det A=\sum_{\pi\in S_n}\operatorname{sgn}(\pi)\prod_{i=1}^n a_{i,\pi(i)}.
$$
Each <product> selects one <coefficient> from each equation and each <unknown>'s column. The <sign of a permutation> determines whether it is added or subtracted. This makes the number $n$ a genuine parameter of a general rule, rather than merely supplying separately memorised formulas for two or three <unknowns>. The notation here is modern, not a transcription of <Alexandre-Théophile Vandermonde>'s symbols.
For two <unknowns>, multiply $ax+by=u$ by $d$ and $cx+dy=v$ by $b$, then subtract. Similarly eliminate $x$. This gives
$$
(ad-bc)x=du-bv,\qquad(ad-bc)y=av-cu.
$$
The same principle in arbitrary size is <Cramer's rule>. If $A_j(\mathbf{b})$ is obtained by replacing column $j$ of $A$ by $\mathbf{b}$, then
$$
\boxed{x_j=\frac{\det A_j(\mathbf{b})}{\det A}\quad\text{when }\det A\ne0.}
$$
An original verification uses the <multilinearity of the determinant>: replace $\mathbf{b}$ by the column combination $\sum_k x_k A_k$. Every resulting <determinant> vanishes through repeated columns except the $k=j$ term, which is $x_j\det A$. This explains how an elimination expression isolates one <unknown>.
The condition $D\ne0$ is substantive. If $D=0$, multiplying by an expression which vanishes cannot justify division. For instance, $x+y=1$ and $2x+2y=3$ are inconsistent, whereas replacing $3$ by $2$ gives infinitely many solutions. Thus an elimination formula must distinguish exceptional <coefficients> and preserve the connection to the original <system of linear equations>. Nor does the displayed rule solve the much broader problem of arbitrary <nonlinear equations>.
The passage's importance is \b[the move from particular elimination calculations towards a theory of alternating expressions]. It would be misleading to turn this into the claim that the familiar <Vandermonde determinant>, formed from powers of variables, is what this extract displays. A positional upper index and an exponent have different meanings; the history of that eponym requires separate evidence. Likewise, modern <matrix algebra> is a useful explanatory language here, not the institutional or conceptual setting of an eighteenth-century memoir.<A case of mathematical eponymy: the Vandermonde determinant by Bernard Ycart>[§§2.1–2.4, separating coefficient indices, difference-products and the later eponym]{ref}
Back to article page
