The Gauss-Jordan Elimination Algorithm

DefinitionsThe AlgorithmSolutions of Linear SystemsAnswering Existence and Uniqueness questionsThe Gauss-Jordan Elimination AlgorithmSolving Systems of Real Linear EquationsA. HavensDepartment of MathematicsUniversity of Massachusetts, AmherstJanuary 24, 2018A. HavensThe Gauss-Jordan Elimination AlgorithmDefinitionsThe AlgorithmSolutions of Linear SystemsAnswering Existence and Uniqueness questionsOutline1 DefinitionsEchelon FormsRow OperationsPivots2 The AlgorithmDescriptionThe Algorithm in practice3 Solutions of Linear SystemsInterpreting RREF of an Augmented MatrixThe 2-variable case: complete solution4 Answering Existence and Uniqueness questionsThe Big QuestionsThree dimensional systemsA. HavensThe Gauss-Jordan Elimination AlgorithmDefinitionsThe AlgorithmSolutions of Linear SystemsAnswering Existence and Uniqueness questionsEchelon FormsRow Echelon FormDefinitionA matrixAis said to be inrow echelon formif the followingconditions hold1all of the rows containing nonzero entries sit above any rowswhose entries are all zero,2the first nonzero entry of any row, called theleading entryofthat row, is positioned to the right of the leading entry of therow above it,Observe: the above properties imply also that all entries of acolumn lying below the leading entry of so

So, one loops over the rst four steps until all pivot columns have been located and pivoting down has occurred in each pivot column. The matrix B k is in row echelon form, with leading 1s in each pivot position. This completes the forward phase. and so the backwards phase commences with, step (5). A. Havens The Gauss-Jordan Elimination Algorithm

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