Linear Algebra with Applications, 10th Edition

1.5 Elementary Matrices

In this section, we view the process of solving a linear system in terms of matrix multiplications rather than row operations. Given a linear system $A\mathbf{x}=\mathbf{b}$, we can multiply both sides by a sequence of special matrices to obtain an equivalent system in row echelon form. The special matrices we will use are called elementary matrices. We will use them to see how to compute the inverse of a nonsingular matrix and also to obtain an important matrix factorization. We begin by considering the effects of multiplying both sides of a linear system by a nonsingular matrix.

Equivalent Systems

Given an $m\times n$ linear system $A\mathbf{x}=\mathbf{b}$, we can obtain an equivalent system by multiplying both sides of the equation by a nonsingular $m\times m$ matrix ...