1. Gauss-Jordan Elimination
[ $A$ | $I$ ] → [ $I$ | $A^{-1}$b]
2. Elementary matrices and a method for finding $A^{-1}$
Def. Matrices A and B are said to be row-equavalent if either (here each) can be obtained from the other by a squence of elementary row operations.
$\widetilde A$ = [ $A$ | $I$ ] → ··· → [ $I$ | $A^{-1}$b]
↘ row-equavalent !
Def. A matrix E is called an elementary matrix if it can be obtained from an identity matrix by performing a single elementary row operation.
[ $A$ | $I$ ] → [ x | E1 ] → [ x | E2 ] → [ x | E3 ] → [ $I$ | $A^{-1}$b]
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