Problem 21 - Intermediate Algebra

Question

Perform each of the following elementary row operations on the augmented matrix \(\left[\begin{array}{rrrr}3 & 6 & -9 & 0 \\ 1 & 5 & -2 & 1 \\ -2 & 2 & -2 & 5\end{array}\right]\) $$ R_{2} \leftrightarrow R_{3} $$

Step-by-Step Solution

Answer

The matrix after performing \( R_2 \leftrightarrow R_3 \) is \( \left[ \begin{array}{rrrr}3 & 6 & -9 & 0 \\ -2 & 2 & -2 & 5 \\ 1 & 5 & -2 & 1 \end{array} \right] \).

Step 1: Identify Row Swap Operation

We need to perform the row operation \( R_2 \leftrightarrow R_3 \), which means we are going to swap row two with row three in the matrix.

Step 2: Write Initial Matrix

The augmented matrix we are starting with is: \[\begin{bmatrix}3 & 6 & -9 & 0 \1 & 5 & -2 & 1 \-2 & 2 & -2 & 5\end{bmatrix}\]

Step 3: Perform Row Swap

According to the operation \( R_2 \leftrightarrow R_3 \), switch the second row with the third row. After the swap, the matrix becomes: \[\begin{bmatrix}3 & 6 & -9 & 0 \-2 & 2 & -2 & 5 \1 & 5 & -2 & 1\end{bmatrix}\]

Step 4: Verify the Result

Double-check the swapped rows to ensure accuracy. The second row \([-2, 2, -2, 5]\) is now in the former position of the third, and the third row \([1, 5, -2, 1]\) is where the second row was, confirming a correct operation.

Key Concepts

Augmented Matrix

An augmented matrix is a tool used to represent a system of linear equations. It combines the coefficients of the variables and the constants from each equation into a single matrix. For example, in the given problem, the augmented matrix is: \[ \begin{bmatrix}3 & 6 & -9 & 0 \ 1 & 5 & -2 & 1 \ -2 & 2 & -2 & 5\end{bmatrix} \]In this matrix, the first three columns represent the coefficients of the variables, while the last column contains the constants from each equation. The structure of an augmented matrix is particularly helpful because it allows us to use matrix operations, such as row swaps and other manipulations, to find solutions to the system of equations involved.

• First three columns: Coefficients of variables.
• Last column: Constants of the equations.
• Represents a full system of linear equations.

By using an augmented matrix, the process of solving systems of linear equations becomes more systematic and organized. This matrix makes it easier to keep track of the operations needed to simplify or solve the equations.

Row Swap Operation

The row swap operation is a simple yet powerful technique used in matrix manipulation, especially when working with an augmented matrix. The operation is denoted as \(R_i \leftrightarrow R_j\), where \(R_i\) and \(R_j\) are the rows being swapped in the matrix. In the original exercise, the operation was given as:\[ R_2 \leftrightarrow R_3 \]This means that we exchange row 2 with row 3. After performing this operation on the given augmented matrix, the new matrix configuration becomes:\[ \begin{bmatrix}3 & 6 & -9 & 0 \ -2 & 2 & -2 & 5 \ 1 & 5 & -2 & 1\end{bmatrix} \]Why would you perform a row swap? Here are some reasons:

• To simplify the matrix and reveal solutions.
• To position pivotal elements in convenient spots.
• To handle unique cases like rearranging equations for clarity.

It is crucial to verify the row swap to ensure accuracy by comparing the initial and resulting matrix configurations. The swap helps in organizing matrices to facilitate easier solution finding and further operations.

Matrix Manipulation

Matrix manipulation involves using various operations to transform a matrix into a desired form. These operations can rearrange, scale, or modify rows and columns to simplify or solve systems of equations. In this context, the main operation involved is the row swap, but there are several others commonly used in conjunction with it:

• Row Addition: Add a multiple of one row to another row.
• Row Scaling: Multiply all entries of a row by a non-zero scalar.
• Row Swap: Exchange two rows in the matrix.

These operations are essential when working to obtain Row Echelon Form (REF) or Reduced Row Echelon Form (RREF) of a matrix. Each operation maintains the equivalence of the matrix, meaning the original system's solutions remain unchanged, allowing you to transform and solve the equations without altering the underlying relationships. By manipulating the matrix, you can expose solutions and make calculations more apparent, helping ensure an accurate and efficient process in solving linear systems.