Kronecker-Capelli theorem

compatibility criterion for a system of linear equations

A system of linear equations $$ \begin{array}{ccc} a_{11} x_1 + \cdots + a_{1n}x_n &=& b_1 \\ \vdots & \vdots & \vdots \\ a_{n1} x_1 + \cdots + a_{nn}x_n &=& b_n \end{array} $$

is compatible if and only if the rank of the coefficient matrix $A = (a_{ij})$ is equal to that of the augmented matrix $\bar A$ obtained from $A$ by adding the column of free terms $b_i$.

Kronecker's version of this theorem is contained in his lectures read at the University of Berlin in 1883–1891 (see [1]). A. Capelli was apparently the first to state the theorem in the above form, using the term "rank of a matrix" (see [2]).

References

[1]	L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903)
[2]	A. Capelli, "Sopra la compatibilitá o incompatibilitá di più equazioni di primo grado fra picì incognite" Revista di Matematica , 2 (1892) pp. 54–58
[3]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)