Matrix (mathematics) - Linear Equations

Linear Equations

Matrices can be used to compactly write and work with multiple linear equations, i.e., systems of linear equations. For example, if A is an m-by-n matrix, x designates a column vector (i.e., n×1-matrix) of n variables x₁, x₂, ..., x_n, and b is an m×1-column vector, then the matrix equation

Ax = b

is equivalent to the system of linear equations

A_1,1x₁ + A_1,2x₂ + ... + A_1,nx_n = b₁

...

A_m,1x₁ + A_m,2x₂ + ... + A_m,nx_n = b_m .