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Elementary Matrix Operations and Elementary Matrices

Left- Multiplication Transformations

Definition

Let A be an m × n matrix. The left multiplication by A is the linear
transformation   defined by

Theorem

Let A and B be n × m matrices. Then


if and only if A = B.
and   for all a ∈ F.
If   is linear , then there exists a unique m × n
matrix C such that

Change of Coordinates for Left -Multiplication
Transformations

Theorem

Let A be an n × n matrix and let
γ be an ordered basis for Fn.
Then where Q is the n × n matrix whose jth
column is the jth vector of
γ.
Definition

Let A be an m × n matrix. Any one of the following three
operations on the rows [columns] of A is called an elementary row
[column] operation :

  interchanging any two rows [columns] of A. (type 1)
  multiplying any row [column] of A by a nonzero scalar. (type 2)
adding any scalar multiple of a row [column] of A to another
row [column]. (type 3)

Elementary Matrices Operations

Definition

An n × n elementary matrix is a matrix obtained by
performing an elementary operation on I n.

The elementary matrix is said to be of type 1, 2, or 3
according to whether the elementary operation performed on
In is a type 1, 2, or 3 operation, respectively.

Multiplying with an Elementary Matrix

Theorem

Let , and suppose that B is obtained from A by
performing an elementary row operation . Then there exists an
m × m elementary matrix such that B = EA. In fact, E is obtained
from Im by performing the same row operation as that which was
performed on A to obtain B.
Conversely, if E is an elementary m × m matrix, then EA is the
matrix obtained from A by performing the same elementary row
operation which produces E from Im.

Every Elementary Matrix is Invertible

Theorem

Elementary matrices are invertible, and the inverse of an elementary
matrix is an elementary matrix of the same type.
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