Elementary Matrix Operations and Elementary Matrices
Left Multiplication Transformations
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 F^{n}.
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
I_{n} 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 I_{m} 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 I_{m}. 
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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