1 / 9

Understanding Elementary Matrices and Inverses: Methods for Finding A−1

This section explores the concept of elementary matrices, which are derived from the identity matrix through single elementary row operations. It outlines the relationship between elementary matrices and row operations in matrix multiplication, demonstrating how performing row operations on a matrix A can be represented by an elementary matrix E. Furthermore, it discusses the invertibility of elementary matrices and presents equivalent statements for an invertible matrix A. A systematic method for finding the inverse of a matrix using elementary row operations is also provided.

vienna
Download Presentation

Understanding Elementary Matrices and Inverses: Methods for Finding A−1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 1.5 Elementary Matrices and a Method for Finding A−1

  2. ELEMENTARY MATRICES An n×n matrix is called an elementary matrix if it can be obtained from the n×n identity matrix In by performing a singled elementary row operation.

  3. ROW OPERATIONS BY MATRIX MULTIPLICATION Theorem 1.5.1: If an elementary matrix E results from performing a certain row operation on Im and if A is an m×n matrix, then the product EA is the matrix that results when the same row operation is performed on A.

  4. INVERSE OPERATIONS Since a single row operation produced the elementary matrix E, there is a row operation , called an inverse operation, that returns E to I.

  5. ELEMENTARY MATRICES AND INVERSES Theorem 1.5.2: Every elementary matrix is invertible, and the inverse is also and elementary matrix.

  6. EQUIVALENT STATEMENTS Theorem 1.5.3: If A is an n×n matrix, then the following statements are equivalent; that is, all are true or all are false. (a) A is invertible. (b) Ax = 0 has only the trivial solution. (c) The reduced row-echelon form of A is In. (d) A is expressible as a product of elementary matrices.

  7. ROW EQUIVALENCY Matrices that can be obtained from one another by a finite sequence of row operations are said to be row equivalent.

  8. A METHOD FOR INVERTING MATRICES To find the inverse of an invertible matrix A, we must find a sequence of elementary row operations that reduces A to the identity matrix and then perform the sequence of row operations on In to obtain A−1.

More Related