Rouché–Capelli Theorem

linear-algebra

Definition

Rouché–Capelli Theorem

The Rouché–Capelli theorem is a solvabilty criterion for linear system of equations.

Let $A\in F^{m\times n}$ and $\mathbf{b}\in F^n$. A linear system of equation $A\cdot \mathbf{x}=\mathbf{b}$ is solvable if and only if:

$$\mathrm{rank}(A)=\mathrm{rank}(A\cdot \mathbf{b})$$

where the matrix $(A\cdot \mathbf{b})$ is called the extended system matrix of the linear system of equation $A\cdot \mathbf{x}=\mathbf{b}$.