Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Fill in the missing entries by performing the indicated row operations to obtain the row-reduced matrices. \(\left[\begin{array}{rrr|r}0 & 1 & 3 & -4 \\ 1 & 2 & 1 & 7 \\ 1 & -2 & 0 & 1\end{array}\right] \frac{R_{1} \leftrightarrow R_{2}}{\longrightarrow}\left[\begin{array}{rrr|r}. & \cdot & \cdot & \cdot \\\ \cdot & \cdot & \cdot & \cdot \\ 1 & -2 & 0 & 1\end{array}\right]\) \(\frac{R_{3}-R_{1}}{\longrightarrow}\left[\begin{array}{rrr|r}1 & 2 & 1 & 7 \\\ 0 & 1 & 3 & -4 \\ \cdot & \cdot & \cdot & \cdot\end{array}\right] \frac{R_{1}+\frac{1}{2} R_{3}}{R_{3}+4 R_{2}}\left[\begin{array}{ccc|r}\cdot & \cdot & \cdot & \cdot \\ 0 & 1 & 3 & -4 \\ \cdot & \cdot & \cdot & \cdot\end{array}\right]\)

Step-by-Step Solution

Answer

After performing the given row operations, we obtain the following row-reduced matrix: \(\left[\begin{array}{rrr|r}1 & 0 & 3/2 & 4 \\\ 0 & 1 & 3 & -4 \\\ 0 & 0 & 11 & -22\end{array}\right]\)

Step 1: Swap rows 1 and 2

First, we swap rows 1 and 2. This is denoted as \(R_1 \leftrightarrow R_2\). Original Matrix: \(\left[\begin{array}{rrr|r}0 & 1 & 3 & -4 \\\ 1 & 2 & 1 & 7 \\\ 1 & -2 & 0 & 1\end{array}\right]\) After swapping rows 1 and 2: \(\left[\begin{array}{rrr|r}1 & 2 & 1 & 7 \\\ 0 & 1 & 3 & -4 \\\ 1 & -2 & 0 & 1\end{array}\right]\)

Step 2: Subtract row 1 from row 3

Now, we perform the second operation: \(R_3 - R_1\). Matrix after step 1: \(\left[\begin{array}{rrr|r}1 & 2 & 1 & 7 \\\ 0 & 1 & 3 & -4 \\\ 1 & -2 & 0 & 1\end{array}\right]\) After subtracting row 1 from row 3: \(\left[\begin{array}{rrr|r}1 & 2 & 1 & 7 \\\ 0 & 1 & 3 & -4 \\\ 0 & -4 & -1 & -6\end{array}\right]\)

Step 3: Add half of row 3 to row 1 and add 4 times row 2 to row 3

Lastly, we perform the two indicated operations simultaneously: \(R_1+\frac{1}{2} R_3\) and \(R_3+4 R_2\). Matrix after step 2: \(\left[\begin{array}{rrr|r}1 & 2 & 1 & 7 \\\ 0 & 1 & 3 & -4 \\\ 0 & -4 & -1 & -6\end{array}\right]\) After performing \(R_1+\frac{1}{2} R_3\) and \(R_3+4 R_2\): \(\left[\begin{array}{rrr|r}1 & 0 & 3/2 & 4 \\\ 0 & 1 & 3 & -4 \\\ 0 & 0 & 11 & -22\end{array}\right]\)

Key Concepts

Row-Reduced Echelon Form

In the pursuit of solutions for systems of linear equations, 'Row-Reduced Echelon Form' (RREF) is a critical milestone. A matrix is in RREF when it meets specific criteria: each leading coefficient (also known as a pivot) is 1, every pivot is to the right of any leading coefficients above it, and pivots are the only nonzero entries in their respective columns. Additionally, any rows consisting entirely of zeros are at the bottom of the matrix.

RREF is convenient because it reveals a lot about the system of equations it represents, including whether the system has no solution, exactly one solution, or infinitely many solutions. By following standardized steps in matrix row operations, we move toward this echelon form. While not every matrix can be reduced to the RREF manually or with a simple calculator, most computational software has built-in functions to compute the RREF.

Systems of Linear Equations

A 'system of linear equations' consists of two or more linear equations containing two or more variables. The objective is to find the values for the variables that satisfy all equations in the system simultaneously. Systems can be solved by various methods, including graphing, substitution, elimination, and matrix approaches - such as Gaussian elimination.

By utilizing matrices, we can represent a system compactly and perform operations that correspond to the aforementioned methods (like elimination), but more systematically. When we fully reduce a matrix to RREF as in the exercise above, a clear correspondence between the row operations and the steps of elimination in the equations can be observed. This translates to each row representing an equation, and the column entries indicating the coefficients of the variables in the equations.

Matrix Operations

Matrix operations are essential tools in solving systems of linear equations efficiently. These operations - which include row swapping (denoted as R_i ↔ R_j), row multiplication, and row addition/subtraction (such as R_i + kR_j where k is a scalar) - allow us to manipulate matrices in ways that gradually simplify them into a form from which solutions can be read off directly.

Key Matrix Operations:

• Row Swapping: Interchanging two rows of a matrix.
• Row Multiplication: Multiplying all entries of a row by a non-zero scalar.
• Row Addition/Subtraction: Adding or subtracting rows after optionally multiplying them by a scalar.

These operations preserve the solutions of the system represented by the matrix and are used systematically in methods like Gaussian elimination.

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations by transforming the system's matrix into an upper triangular form or into row-reduced echelon form (RREF), which can then be solved by back-substitution. It employs sequential matrix operations to clear columns below the leading entry, which corresponds to the elimination of variables in the equations.

The process involves a series of steps: identifying the pivot, using row operations to create zeros below the pivot, moving to the next pivot to the right and below the current one, and repeating the process until the desired form is achieved. As we see in the text exercise, these steps are executed in sequence, with each step building on the last to simplify the matrix systematically. This is what makes Gaussian elimination a cornerstone technique in linear algebra and crucial for understanding multivariable systems.