Sec 3.5 Inverses of Matrices Where A is nxn Finding the inverse of A: Seq or row operations

Finding the inverse of A: A Find inverse

Properties

Fact1: AB in terms of columns of B Fact1: Ax in terms of columns of A

Basic unit vector: J-th location

Def: A is invertable if There exists a matrix B such that TH1: the invers is unique TH2: the invers of 2x2 matrix Find inverse

TH3: Algebra of inverse If A and B are invertible, then

TH4: solution of Ax = b Solve

Def: E is elementary matrix if 1) Square matrix nxn 2) Obtained from I by a single row operation

Which of these matrices are elementary matrices

REMARK: Let E corresponds to a certain elem row operation. It turns out that if we perform this same operation on matrix A, we get the product matrix EA

NOTE: Every elementary matrix is invertible

Sec 3.5 Inverses of Matrices TH6: A is row equivalent to identity matrix I Row operation 1Row operation 2Row operation 3Row operation k A is invertible A is invertable A is a product of elementary matrices

Solve Solving linear system Solve What is the solution

Quiz2: SAT in Class ( ) 1) Given a matrix A find the reduced row echelon form 2) Use the method of Gauss-Jordan elimination to solve the following system (find the solution in vector form (i.e) as a linear combination of vectors) Quiz3: Sund online ( )

Solve Matrix Equation In certain applications, one need to solve a system Ax = b of n equations in n unknowns several times but with different vectors b1, b2,.. Matrix Equation

Definition: A is nonsingular matrix if the system has only the trivial solution Show that A is nonsingular RECALL: Definitions invertible Row equivalent nonsingular

Theorem7:(p193) row equivalentnonsingular Ax = b Every n-vector b has unique sol Ax = b Every n-vector b is consistent Ax = 0 The system has only the trivial sol is a product of elementary matrices

TH7: A is an nxn matrix. The following is equivalent (a) A is invertible (b) A is row equivalent to the nxn identity matrix I (c) Ax = 0 has the trivial solution (d) For every n-vector b, the system A x = b has a unique solution (e) For every n-vector b, the system A x = b is consistent

Problems (page194) 34) Show that a diagonal matrix is inverible if and only if each diagonal element is nonzero. In this case, state concisely how the invers matrix is obtained. 35) Let A be an nxn matrix with either a row or a column consisting only of zeros. Show that A is not invertible. 41) Show that the i-th row of the product AB is A i B, where A i is the i-th row of the matrix A.

True & False row equivalentnonsingular Ax = b Every n-vector b has unique sol Ax = b Every n-vector b is consistent Ax = 0 The system has only the trivial sol is a product of elementary matrices ? ? ? ? ? ?