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

Finding the inverse of 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

What is The Big Day Register ???

TH1: the invers is unique 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

If A and B are invertible, then TH3: Algebra of inverse If A and B are invertible, then 1 2 3 4

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

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 invertible if and only if it is row equivalent to identity matrix I Row operation 1 Row operation 2 Row operation 3 Row operation k

Solving linear system Solve

Matrix Equation Solve

Definition: A is nonsingular matrix if the system has only the trivial solution

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