Problem 123 - Precalculus with Limits

Question

What is the relationship between the three elementary row operations performed on an augmented matrix and the operations that lead to equivalent systems of equations?

Step-by-Step Solution

Answer

The three elementary row operations on an augmented matrix correspond to logical operations on the system of equations, leading to an equivalent system. These operations include swapping the order of equations (equivalent to swapping two rows), multiplying an equation by a non-zero scalar (equivalent to scaling a row), and adding or subtracting equations (equivalent to adding or subtracting rows).

Step 1: Elementary Row Operation 1 - Swapping Rows

This operation involves swapping two rows in the matrix. This corresponds to swapping the order of equations in the system of equations. Note, the order of equations doesn't affect the solutions of the system. So, if the i-th and j-th rows are switched in the matrix, it is equivalent to switching the i-th and j-th equations in the system.

Step 2: Elementary Row Operation 2 - Scaling a Row

This operation involves multiplying a row by non-zero scalar. This corresponds to multiplying every term in an equation by the same non-zero scalar. If a row in the matrix is multiplied by a scalar, it means the corresponding equation in the system is also multiplied by the same scalar. The solutions to the system remain the same because every term of the equation changes by the same factor.

Step 3: Elementary Row Operation 3 - Adding or Subtracting Rows

This operation involves adding or subtracting a multiple of one row to/from another row. This corresponds to adding or subtracting equations in the system. When one row is added to or subtracted from another in the matrix, it is equivalent to adding or subtracting the corresponding equations in the system. This doesn't affect the solutions of the system as it is creating a new equation that is still true for the known solutions.