Linear Equations by Rank Method In section . . , we have already defined consistency of a system of linear equation. In this section, we investigate it by using rank method.
We state the following theorem without proof: Theorem . (Rouché - Capelli Theorem) A system of linear equations, written in the matrix form as AX B , is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix; that is, ρ ρ ( ) ([ ]). A B We apply the theorem in the following examples.