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Inverse of Matrix. Gauss-Jordan Elimination Part 5. Fundamentals of Engineering Analysis. Eng. Hassan S. Migdadi. Inverses of Matrices. Where A is nxn. Finding the inverse of A:. Seq or row operations. Finding the inverse of A:. A. Find inverse. Def: A is invertable if.

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## Fundamentals of Engineering Analysis

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**Inverse of Matrix. Gauss-Jordan Elimination**Part 5 Fundamentals of Engineering Analysis Eng. Hassan S. Migdadi**Inverses of Matrices**Where A is nxn Finding the inverse of A: Seq or row operations**Finding the inverse of A:**A Find inverse**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 1 2 3 4**TH4: solution of Ax = b**Solve**Solving linear system**Solve Solve What is the solution**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**Definition:**A is nonsingular matrix if the system has only the trivial solution RECALL: Definitions Show that A is nonsingular invertible Row equivalent nonsingular**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**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**?**? 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

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