3.148 min read

Solvability: When Does Ax = b Have a Solution?

The system $A\mathbf{x} = \mathbf{b}$ has a solution if and only if $\mathbf{b}$ lies in the column space of $A$ . Operationally, we check this by row-reducing the augmented matrix $[A | \mathbf{b}]$ and looking for contradictory rows.

A contradictory row $[0, 0, \ldots, 0 | c]$ with $c \neq 0$ signals that $\mathbf{b}$ is NOT in $\text{Col}(A)$ — the system is inconsistent.

If no contradiction arises, $\mathbf{b} \in \text{Col}(A)$ and the system is consistent. The solution is found by back substitution, with free variables parametrizing the null space.

Formal View

Theorem 3.14

The augmented system

[A | \mathbf{b}]

row-reduces to a system with no contradictory row

\Longleftrightarrow

\mathbf{b} \in \text{Col}(A)

\Longleftrightarrow

A\mathbf{x} = \mathbf{b}

is consistent.

Why This Matters

The solvability check is the first step in any practical linear algebra computation.

Before running any solver, verify the system is consistent
In optimization, infeasibility detection saves computational effort

Learning Resources

Consistency and solvability of linear systems

Khan Academy

Augmented matrix and solvability criterion.

13 min

Quiz

Question 1

During row reduction of $[A|\mathbf{b}]$ you get the row $[0, 0, 0 | 3]$ . The system is:

Common Mistakes

Stopping row reduction before checking all rows for contradictions.
Assuming a system with many equations is more likely to be solvable — more constraints can only reduce solvability.