54. Linear System Solutions

A linear system $𝐴\overrightarrow{𝑥}=\overrightarrow{𝑏}$ has exactly one of three outcomes:

1. Unique solution — exactly one $\overrightarrow{𝑥}$ satisfies the system
2. No solution — the system is inconsistent
3. Infinitely many solutions — the system is underdetermined

The REF (or RREF) of the augmented matrix $\left[𝐴|\overrightarrow{𝑏}\right]$ tells you which case you’re in.

54.1. 1. Unique solution

Every column of $𝐴$ has a pivot, no free variables, and no inconsistency:

Equivalently: $\text{rank}\left(𝐴\right)=\text{rank}\left(\left[𝐴|\overrightarrow{𝑏}\right]\right)=𝑛$ (the number of unknowns).

54.2. 2. No solution (inconsistent)

A row of the form $\left[0,0,\dots ,0|𝑐\right]$ with $𝑐\ne 0$ — translates to “$0=𝑐$” which is impossible.

The second row says $0𝑥+0𝑦=1$. No $\overrightarrow{𝑥}$ satisfies this.

Equivalently: $\text{rank}\left(𝐴\right)<\text{rank}\left(\left[𝐴|\overrightarrow{𝑏}\right]\right)$.

54.3. 3. Infinitely many solutions (free variables)

Some columns of $𝐴$ have no pivot — those correspond to free variables you can set arbitrarily; the pivot variables are determined by them.

Columns 2 and 4 have no pivot → ${𝑥}_2,{𝑥}_4$ are free.

Equivalently: $\text{rank}\left(𝐴\right)=\text{rank}\left(\left[𝐴|\overrightarrow{𝑏}\right]\right)<𝑛$. The solution set is an affine subspace of dimension $𝑛-\text{rank}\left(𝐴\right)$ — see Rank–Nullity.

54.4. Rouché–Capelli summary

54.5. See also

• Row Echelon Form — the algorithm
• Linear System Special Cases — when zero rows appear
• Rank and Rank–Nullity
• Homogeneous System — special case $\overrightarrow{𝑏}=\mathrm{𝟎}$