{"id":2467,"date":"2024-07-29T23:59:28","date_gmt":"2024-07-29T23:59:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2467"},"modified":"2025-08-15T16:28:20","modified_gmt":"2025-08-15T16:28:20","slug":"solving-system-of-equations-using-matrices-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/solving-system-of-equations-using-matrices-learn-it-2\/","title":{"raw":"Solving System of Equations using Matrices: Learn It 2","rendered":"Solving System of Equations using Matrices: Learn It 2"},"content":{"raw":"

Row Operations<\/h2>\r\nNow that we can write systems of equations in augmented matrix form, we will examine the various\u00a0row operations<\/strong> that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows. Performing row operations on a matrix is the method we use for solving a system of equations.\r\n\r\nWhen solving systems of equations using matrices, our goal is to transform the matrix into a simpler form while maintaining the same solution. Row operations help us do this systematically. Think of it like simplifying an equation - we can perform certain operations that don't change the solution but make it easier to solve.\r\n\r\nIn a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.\r\n
    \r\n \t
  1. Interchange any two rows.<\/li>\r\n \t
  2. Multiply a row by any real number except [latex]0[\/latex].<\/li>\r\n \t
  3. Add a nonzero multiple of one row to another row.<\/li>\r\n<\/ol>\r\nThese actions are called row operations and will help us use the matrix to solve a system of equations.\r\n

    When we perform row operations, we use special notation to show what we're doing. Here are the common notations you'll see:<\/p>\r\n

    For swapping rows:<\/p>\r\n\r\n

      \r\n \t
    • [latex]R_i \\leftrightarrow R_j[\/latex] means we swap row [latex]i[\/latex] and row [latex]j[\/latex]<\/li>\r\n<\/ul>\r\n

      For multiplying a row by a constant:<\/p>\r\n\r\n

        \r\n \t
      • [latex]cR_i[\/latex] or [latex]R_i \\rightarrow cR_i[\/latex] means we multiply row [latex]i[\/latex] by the constant [latex]c[\/latex]<\/li>\r\n \t
      • Example: [latex]5R_2 \\rightarrow R_2[\/latex] means multiply row [latex]2[\/latex] by [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n

        For adding multiples of rows:<\/p>\r\n\r\n

          \r\n \t
        • [latex]cR_i + R_j \\rightarrow R_j[\/latex] means we multiply row [latex]i[\/latex] by [latex]c[\/latex] and add it to row [latex]j[\/latex]<\/li>\r\n \t
        • [latex]R_j \\rightarrow R_j + cR_i[\/latex] is another common way to write this<\/li>\r\n \t
        • Example: [latex]-3R_3 + R_1 \\rightarrow R_1[\/latex] means multiply row [latex]3[\/latex] by [latex]-3[\/latex] and add to row [latex]1[\/latex]<\/li>\r\n<\/ul>\r\n

          These operations can be combined and will be used throughout our work with matrices to solve systems of equations.<\/p>\r\n\r\n

          Use the indicated row operations on the augmented matrix:[latex]\\left[\\begin{array}{ccc|c}\\hfill 6& \\hfill -5& \\hfill 2& \\hfill 3\\\\ \\hfill 1& \\hfill 1& \\hfill -4& \\hfill 5\\\\ \\hfill 3& \\hfill -3& \\hfill 1& \\hfill -1\\\\ \\end{array}\\right][\/latex]\r\n
            \r\n \t
          1. Interchange rows [latex]2[\/latex] and [latex]3[\/latex].\r\n[reveal-answer q=\"309285\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"309285\"]\r\n\r\n[caption id=\"attachment_2468\" align=\"aligncenter\" width=\"768\"]\"\" Interchanging rows 2 and 3[\/caption]\r\n\r\n[\/hidden-answer]<\/li>\r\n \t
          2. Multiply row [latex]2[\/latex] by [latex]5[\/latex].\r\n[reveal-answer q=\"623293\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"623293\"]\r\n\r\n[caption id=\"attachment_2469\" align=\"aligncenter\" width=\"664\"]\"\" Multiplying row 2 by 5[\/caption]\r\n\r\n[\/hidden-answer]<\/li>\r\n \t
          3. Multiply row [latex]3[\/latex] by [latex]\u22122[\/latex] and add to row [latex]1[\/latex].\r\n[reveal-answer q=\"290247\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"290247\"]<\/span><\/span>\r\n\r\n[caption id=\"attachment_2470\" align=\"aligncenter\" width=\"1244\"]\"\" Finalizing row operations to isolate a variable[\/caption]\r\n\r\n[\/hidden-answer]<\/span><\/li>\r\n<\/ol>\r\n<\/section>
            [ohm2_question hide_question_numbers=1]24847[\/ohm2_question]<\/section>
            [ohm2_question hide_question_numbers=1]24848[\/ohm2_question]<\/section>","rendered":"

            Row Operations<\/h2>\n

            Now that we can write systems of equations in augmented matrix form, we will examine the various\u00a0row operations<\/strong> that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows. Performing row operations on a matrix is the method we use for solving a system of equations.<\/p>\n

            When solving systems of equations using matrices, our goal is to transform the matrix into a simpler form while maintaining the same solution. Row operations help us do this systematically. Think of it like simplifying an equation – we can perform certain operations that don’t change the solution but make it easier to solve.<\/p>\n

            In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.<\/p>\n

              \n
            1. Interchange any two rows.<\/li>\n
            2. Multiply a row by any real number except [latex]0[\/latex].<\/li>\n
            3. Add a nonzero multiple of one row to another row.<\/li>\n<\/ol>\n

              These actions are called row operations and will help us use the matrix to solve a system of equations.<\/p>\n

              When we perform row operations, we use special notation to show what we’re doing. Here are the common notations you’ll see:<\/p>\n

              For swapping rows:<\/p>\n

                \n
              • [latex]R_i \\leftrightarrow R_j[\/latex] means we swap row [latex]i[\/latex] and row [latex]j[\/latex]<\/li>\n<\/ul>\n

                For multiplying a row by a constant:<\/p>\n

                  \n
                • [latex]cR_i[\/latex] or [latex]R_i \\rightarrow cR_i[\/latex] means we multiply row [latex]i[\/latex] by the constant [latex]c[\/latex]<\/li>\n
                • Example: [latex]5R_2 \\rightarrow R_2[\/latex] means multiply row [latex]2[\/latex] by [latex]5[\/latex]<\/li>\n<\/ul>\n

                  For adding multiples of rows:<\/p>\n

                    \n
                  • [latex]cR_i + R_j \\rightarrow R_j[\/latex] means we multiply row [latex]i[\/latex] by [latex]c[\/latex] and add it to row [latex]j[\/latex]<\/li>\n
                  • [latex]R_j \\rightarrow R_j + cR_i[\/latex] is another common way to write this<\/li>\n
                  • Example: [latex]-3R_3 + R_1 \\rightarrow R_1[\/latex] means multiply row [latex]3[\/latex] by [latex]-3[\/latex] and add to row [latex]1[\/latex]<\/li>\n<\/ul>\n

                    These operations can be combined and will be used throughout our work with matrices to solve systems of equations.<\/p>\n

                    Use the indicated row operations on the augmented matrix:[latex]\\left[\\begin{array}{ccc|c}\\hfill 6& \\hfill -5& \\hfill 2& \\hfill 3\\\\ \\hfill 1& \\hfill 1& \\hfill -4& \\hfill 5\\\\ \\hfill 3& \\hfill -3& \\hfill 1& \\hfill -1\\\\ \\end{array}\\right][\/latex]<\/p>\n
                      \n
                    1. Interchange rows [latex]2[\/latex] and [latex]3[\/latex].\n