A linear system is equivalent to a matrix equation.

a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂
...
aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ

[a₁₁ a₁₂ ... a₁ₙ] [x₁]   [b₁]
[a₂₁ a₂₂ ... a₂ₙ] [x₂] = [b₂]
[...  ...  ... ...] [...]  [...]
[aₘ₁ aₘ₂ ... aₘₙ] [xₙ]   [bₘ]

**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.Copy