**Equivalence of Linear Systems and Matrix Equations**
A system of linear equations can be expressed compactly as a matrix equation `Ax = b`, where:
- `A` is the `m × n` coefficient matrix
- `x` is the `n × 1` column vector of unknowns
- `b` is the `m × 1` column vector of constants
For the system:
```
a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂
...
aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ
```
The matrix form is:
```
[a₁₁ a₁₂ ... a₁ₙ] [x₁] [b₁]
[a₂₁ a₂₂ ... a₂ₙ] [x₂] = [b₂]
[... ... ... ...] [...] [...]
[aₘ₁ aₘ₂ ... aₘₙ] [xₙ] [bₘ]
```
This compact notation enables the use of matrix algebra techniques (row reduction, inverses, determinants) to solve systems efficiently.