Linear System Special Cases

55. Linear System Special Cases

When you reduce an augmented matrix $[𝐴 | \vec{𝑏}]$ to REF, rows of all zeros (or zero coefficients with non-zero RHS) signal one of three structural situations.

55.1. 1. Dependent equations

One equation is a linear combination of others — they carry the same information.

\begin{matrix} 2 𝑥 + 4 𝑦 = 8 \\ 𝑥 + 2 𝑦 = 4 \end{matrix}

The second equation is half the first — same line. After row reduction:

[\begin{matrix} 1 & 2 & 4 \\ 0 & 0 & 0 \end{matrix}]

The zero row means one equation was redundant. The system has infinitely many solutions (a whole line in $ℝ^{2}$ ).

55.2. 2. Underdetermined systems

More variables than independent equations — fewer constraints than unknowns.

\begin{matrix} 𝑥 + 𝑦 + 𝑧 = 2 \\ 2 𝑥 + 3 𝑦 + 𝑧 = 5 \end{matrix}

Two equations in three unknowns. After row reduction:

[\begin{matrix} 1 & 1 & 1 & 2 \\ 0 & 1 & - 1 & 1 \end{matrix}]

Both equations are independent, but $𝑧$ is a free variable. Infinitely many solutions (a line in $ℝ^{3}$ ).

55.3. 3. Inconsistent systems

A zero coefficient row with a non-zero RHS — translates to $0 = 𝑐$ for $𝑐 \neq 0$ , which is impossible.

\begin{matrix} 𝑥 + 𝑦 = 3 \\ 2 𝑥 + 2 𝑦 = 7 \end{matrix}

(Same left-hand side scaled by $2$ , but right-hand side doesn’t scale to match.)

[\begin{matrix} 1 & 1 & 3 \\ 0 & 0 & 1 \end{matrix}]

The second row reads $0 = 1$ . No solution.

55.4. See also

Row Echelon Form
Linear System Solutions — Rouché–Capelli summary
Rank — diagnoses dependence