1. From Systems of Equations to Matrix Form
The Upgrade: Why Matrices?
You already know how to solve systems of two or three linear equations by substitution or elimination. Now we compress those systems into a single matrix equation. This isn't just shorthand — it unlocks a uniform theory for systems of any size and lets us bring the full machinery of determinants, inverses, and rank to bear.
Recall: A System of Two Equations
Consider the familiar system:
$$\begin{cases} a_{1}x + b_{1}y = c_{1} \\ a_{2}x + b_{2}y = c_{2} \end{cases}$$We separate the coefficients, the unknowns, and the right-hand side:
$$\underbrace{\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}}_{A}\;\underbrace{\begin{pmatrix} x \\ y \end{pmatrix}}_{\mathbf{x}} = \underbrace{\begin{pmatrix} c_1 \\ c_2 \end{pmatrix}}_{\mathbf{b}}$$A System of Three Equations
$$\begin{cases} a_{1}x + b_{1}y + c_{1}z = d_{1} \\ a_{2}x + b_{2}y + c_{2}z = d_{2} \\ a_{3}x + b_{3}y + c_{3}z = d_{3} \end{cases} \quad\Longrightarrow\quad \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}$$Key Definitions
| Name | Symbol | Description |
|---|---|---|
| Coefficient matrix | $A$ | The $m \times n$ matrix of coefficients of the unknowns |
| Unknown column | $\mathbf{x}$ | The $n \times 1$ column vector of variables |
| Right-hand column | $\mathbf{b}$ | The $m \times 1$ column of constants |
| Augmented matrix | $[A \mid \mathbf{b}]$ | $A$ with $\mathbf{b}$ appended as an extra column, separated by a vertical line |
The compact equation is simply:
$$A\mathbf{x} = \mathbf{b}$$What is gained? Every system — whether $2 \times 2$, $3 \times 3$, or $100 \times 47$ — is now written in the same form. All the tools from matrix algebra (row operations, determinants, inverses, rank) apply uniformly.