A linear system is equivalent to a matrix equation.

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.