A matrix equation is equivalent to a linear system

Theorem: Matrix Equation and Linear System Equivalence

A matrix equation Ax = b is equivalent to the corresponding system of linear equations. Every solution of the matrix equation is a solution of the linear system, and vice versa.

Statement: Given an m × n matrix A, the matrix equation Ax = b represents exactly the same mathematical object as the linear system:

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

Why: By the definition of matrix-vector multiplication, Ax produces an m × 1 vector whose i-th entry is the dot product of row i of A with x. Setting this equal to b entry-by-entry recovers the original system.

Implication: Any technique for solving linear systems (row reduction, Gaussian elimination) applies directly to matrix equations, and any property of matrix equations (existence, uniqueness) translates to properties of linear systems.