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4. Matrices and Row-Equivalent Operations Gaussian Elimination Gauss-Jordan Elimination Matrices Systems of Equations and Matrices > Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_43&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. Because elementary row operations are reversible, row equivalence is an equivalence relation. An elementary row operation is any one of the following moves: Swap (swap two rows of a matrix), Scale (multiply a row of a matrix by a nonzero constant), or Pivot (add to one row of a matrix some multiple of another row). If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Matrices and Row-Equivalent Operations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrices-43/matrices-and-row- equivalent-operations-197- 11208?campaign_content=book_196_section_43&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless A matrix View on Boundless.com Systems of Equations and Matrices > Matrices

6. Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix. There are three types of elementary row operations: swap the positions of two rows, multiply a row by a nonzero scalar, and add to one row a scalar multiple of another. In practice, one does not usually deal with the systems in terms of equations but instead makes use of the augmented matrix (which is also suitable for computer manipulations). Gaussian Elimination Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrices-43/gaussian-elimination-198- 5871?campaign_content=book_196_section_43&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess A matrix View on Boundless.com Systems of Equations and Matrices > Matrices

7. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards. Gauss–Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. Gauss-Jordan elimination, like Gaussian elimination, is used for inverting matrices and solving systems of linear equations. Gauss-Jordan Elimination Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrices-43/gauss-jordan-elimination- 199- 1093?campaign_content=book_196_section_43&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Matrix in Reduced Row Echelon View on Boundless.com Systems of Equations and Matrices > Matrices

8. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

9. Key terms augmented matrix In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Gauss-Jordan elimination In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is a variation of Gaussian elimination. row equivalent In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

10. Matrix in Reduced Row Echelon A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Reduced row echelon form." GNU FDL http://en.wikipedia.org/wiki/Reduced_row_echelon_form#Reduced_row_echelon_form View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Reduced_row_echelon_form#Reduced_row_echelon_formView on Boundless.com Systems of Equations and Matrices

11. A matrix Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Matrix (mathematics)." GNU FDL http://en.wikipedia.org/wiki/Matrix_(mathematics) View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Matrix_(mathematics)View on Boundless.com Systems of Equations and Matrices

12. A matrix Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Matrix (mathematics)." GNU FDL http://en.wikipedia.org/wiki/Matrix_(mathematics) View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Matrix_(mathematics)View on Boundless.com Systems of Equations and Matrices

13. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Which of the following represents the proper order one must take when operating with elementary row operations? A) The order does not matter. B) Pivot, scale, swap. C) Swap, scale, pivot. D) Scale, pivot, swap.

14. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Which of the following represents the proper order one must take when operating with elementary row operations? A) The order does not matter. B) Pivot, scale, swap. C) Swap, scale, pivot. D) Scale, pivot, swap.

15. Attribution Wikipedia. "Gauss–Jordan elimination." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Gauss%25E2%2580%2593Jordan_eliminationCC BY-SA 3.0 http://en.wikipedia.org/wiki/Gauss%25E2%2580%2593Jordan_elimination Wikipedia. "Gauss-Jordan elimination." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Gauss-Jordan%20eliminationCC BY-SA 3.0http://en.wikipedia.org/wiki/Gauss-Jordan%20elimination Wikipedia. "Row echelon form." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Row_echelon_formCC BY-SA 3.0http://en.wikipedia.org/wiki/Row_echelon_form Wikipedia. "System of linear equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_linear_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_linear_equations Wikipedia. "Augmented matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Augmented_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Augmented_matrix Connexions. "Row Reduced Form." CC BY 3.0 http://cnx.org/content/m10295/latest/CC BY 3.0http://cnx.org/content/m10295/latest/ Wikipedia. "Gaussian elimination." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Gaussian_eliminationCC BY-SA 3.0http://en.wikipedia.org/wiki/Gaussian_elimination Wikipedia. "Triangular matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Triangular_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Triangular_matrix Wikipedia. "Elementary row operations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Elementary_row_operationsCC BY-SA 3.0http://en.wikipedia.org/wiki/Elementary_row_operations Wikipedia. "augmented matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/augmented%20matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/augmented%20matrix Wikipedia. "Matrix (mathematics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Matrix_(mathematics)CC BY-SA 3.0http://en.wikipedia.org/wiki/Matrix_(mathematics) Wikipedia. "Row equivalence." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Row_equivalenceCC BY-SA 3.0http://en.wikipedia.org/wiki/Row_equivalence Wikipedia. "row equivalent." CC BY-SA 3.0 http://en.wikipedia.org/wiki/row%20equivalentCC BY-SA 3.0http://en.wikipedia.org/wiki/row%20equivalent Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices