Let A be an n × n matrix and let γ be an ordered basis for Fn.
where Q is the n × n matrix whose jth
column is the jth vector of γ.
Let A be an m × n matrix. Any one of the following three
ope rations on the rows [columns] of A is called an elementary row
interchanging any two rows
[columns] of A. (type 1)
multiplying any row [column]
of A by a non zero scalar . (type 2) adding any scalar multiple of a row [column] of A to another
row [column]. (type 3)
Elementary Matrices Operations
An n × n elementary matrix is a matrix obtained by
performing an elementary operation on In.
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
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
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
Elementary matrices are invertible, and the inverse of an elementary
matrix is an elementary matrix of the same type.
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