What are the matrix method of linear systems?

Consider the linear system of equations AX = Y

where

X is a n x 1 matrix of variables,

Y is a n x 1 matrix of constants, and

A is an n x n matrix of coefficients.

Provided A is not a singular matrix, A has an inverse, A-1, an n x n matrix.

Premultiplying by A-1 gives A-1AX = A-1Y or X = A-1Y, the solution to the linear system.