Invertible
An overview of invertible matrices, including properties, definitions, and implications. Discussions on invertibility, one-to-one and onto functions, matrix characteristics, and related concepts. Explore the significance of invertibility through matrices and functions in mathematics.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Summary Let A be an n x n matrix. A is invertible if and only if The columns of A span Rn For every b in Rn, the system Ax=b is consistent The rank of A is n The columns of A are linear independent The only solution to Ax=0 is the zero vector The nullity of A is zero The reduced row echelon form of A is In A is a product of elementary matrices There exists an n x n matrix B such that BA = In There exists an n x n matrix C such that AC = In
Review - Terminology What actually come out of function f Range ( ) Given a function f ?1 ? ?1 ?2 ? ?2 = ? ?3 ?3 Co-domain ( ) Domain ( ) What can go into function f What may possible come out of function f
Review - Terminology one-to-one ( ) Onto ( ) ?1 ?1 ? ?1 ? ?1 ?2 ?2 ? ?2 = ? ?3 ? ?2 ?3 ? ?3 ?3 Co-domain = range
Review: One-to-one 2 x 3 A function f is one-to-one If co-domain is smaller than the domain, f cannot be one-to-one. ?1 ? ?1 If a matrix A is , it cannot be one-to-one. ?2 ? ?2 ? ?3 ?3 The reverse is not true. ? If a matrix A is one-to- one, its columns are independent. ? ? = ? has one solution ? ? = ? has at most one solution
Review: Onto 3 x 2 A function f is onto If co-domain is larger than the domain, f cannot be onto. ?1 ? ?1 If a matrix A is , it cannot be onto. ?2 ? ?2 = ? ?3 ?3 The reverse is not true. If a matrix A is onto, rank A = no. of rows. Co-domain = range ? ? = ? always have solution
Invertible A is called invertible if there is a matrix B such that ?? = ? and ?? = ? (? = ? 1) ? ? ? 1? ?? ? ? ? 1 ? 1 A must be onto A must be one-to-one ( ? 1 input )
An invertible matrix A is always square. One-to-one and onto A function f is one-to-one and onto The domain and co- domain must have the same size . The corresponding matrix A is square. ?1 ? ?1 ?2 ? ?2 ? ?3 ?3 Onto One-to-one Square
Summary Let A be an n x n matrix. A is invertible if and only if The columns of A span Rn For every b in Rn, the system Ax=b is consistent The rank of A is n The columns of A are linear independent The only solution to Ax=0 is the zero vector The nullity of A is zero The reduced row echelon form of A is In A is a product of elementary matrices There exists an n x n matrix B such that BA = In There exists an n x n matrix C such that AC = In
