Fundamentals of Engineering Analysis

1. Inverse of Matrix. Gauss-Jordan Elimination Part 5 Fundamentals of Engineering Analysis Eng. Hassan S. Migdadi

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

3. Finding the inverse of A: A Find inverse

4. 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

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

6. TH4: solution of Ax = b Solve

7. Solving linear system Solve Solve What is the solution

8. 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,.. Solve Matrix Equation

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

10. Theorem7:(p193) row equivalent nonsingular is a product of elementary matrices Every n-vector b The system Every n-vector b Ax = b Ax = 0 Ax = b has unique sol has only the trivial sol is consistent All statements are equivalent

11. 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

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