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

2. Finding the inverse of A:
Find inverse

3. Properties

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

5. Basic unit vector:
J-th location

6. What is The Big Day
Register ???

7. 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

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

9. TH4: solution of Ax = b
Solve

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

11. 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

12. NOTE:
Every elementary matrix is invertible

13. 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

14. Solving linear system
Solve

15. Matrix Equation
Solve

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

17. 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