Systems of Linear Equations and Gaussian Elimination

A system of linear equations in matrix form:

$$A\vec{x} = \vec{b}$$

Gaussian elimination reduces the augmented matrix $[A|\vec{b}]$ to row echelon form:

1. Swap two rows: $R_i \leftrightarrow R_j$.
2. Scale a row: $R_i \to \alpha R_i$, $\alpha \neq 0$.
3. Add a multiple: $R_i \to R_i + \alpha R_j$.

Example — Solve the $2\times 2$ system:

$$\begin{pmatrix} 2 & 1 & | & 5 \\ 4 & -1 & | & 1 \end{pmatrix} \xrightarrow{R_2 - 2R_1} \begin{pmatrix} 2 & 1 & | & 5 \\ 0 & -3 & | & -9 \end{pmatrix}$$

Back-substitution gives $y = 3$, $x = 1$.

Solution types

• Unique: $\mathrm{rk}(A) = \mathrm{rk}([A|b]) = n$
• Infinite: $\mathrm{rk}(A) = \mathrm{rk}([A|b]) < n$
• None: $\mathrm{rk}(A) < \mathrm{rk}([A|b])$

Tip: Always check the rank of both $A$ and $[A|\vec{b}]$ before interpreting the solution.

Common pitfall: Row operations preserve the solution set, not the matrix: the echelon form is a different matrix telling the same story. And a row $0\;0\;|\;c$ with $c \ne 0$ ends that story — no solution, stop.