Rouché–Capelli theorem

Statement

Let $F$ be a field, let $A$ be an $m \times n$ matrix over $F$, and let $b \in F^m$. The system $Ax = b$ has a solution if and only if $\rank A = \rank [A \mid b]$, where $[A \mid b]$ is the augmented matrix and $\rank$ is the rank of a matrix. If a solution exists, then the solution set is an affine subspace of $F^n$ of dimension $n - \rank A$. In particular, the solution is unique if and only if $\rank A = n$.

Remarks

The theorem is also called the Kronecker–Capelli theorem. If $\rank [A \mid b] = \rank A + 1$, then the system is inconsistent. The dimension $n - \rank A$ is the number of free variables.

Explanations

L1 human

The theorem says that a system has a solution exactly when the right side adds no new independent information. If adding the column $b$ raises the rank, then $b$ is not a combination of the columns of $A$, and no solution exists. When solutions exist, each unknown without a pivot is free, and it adds one dimension to the solution set.