Applications of Matrices: Consistency of System of Linear Equations by Rank Method

The three methods so far (matrix inversion, Cramer's rule, and to a lesser extent Gaussian elimination applied only to a unique-solution case) leave one question unanswered in general: given any system $AX=B$ (square or not, singular or not), how do we decide -- without guessing -- whether it is consistent at all, and if so how many solutions it has? The answer is the rank of the coefficient matrix versus the rank of the augmented matrix.

Theorem 1.14 (Rouché-Capelli Theorem). The system $AX=B$ is consistent if and only if

$\rho(A)=\rho([A\,|\,B]).$ …