Systems of Linear Equations and Gaussian Elimination

Linear Systems & Gaussian Form

A system of linear equations in matrix form is written as $A\vec{x} = \vec{b}$, where $A$ is the coefficient matrix, $\vec{x}$ is the variable vector, and $\vec{b}$ is the constant vector.

Gaussian elimination solves this by reducing the augmented matrix $[A|\vec{b}]$ into row echelon form using three safe operations:

Example: Solve a $2\times 2$ 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 yields $y = 3$ and $x = 1$. Row operations preserve the solution set, but yield a different matrix telling the same story.

Solution Types & Ranks

Always check the rank of both $A$ and the augmented matrix $[A|\vec{b}]$ before interpreting your final system outcome.

Common pitfall: A resulting row of the form $0\;0\;|\;c$ where $c \neq 0$ instantly means no solution. Stop immediately when this contradiction appears.