  # 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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