A matrix equation is equivalent to a linear system

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

**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.Copy