Problem 32 - Precalculus

Question

Perform the indicated row operations on each augmented matrix. $$\left[\begin{array}{rr|r} 2 & -3 & -4 \\ 1 & 2 & 5 \end{array}\right] \quad R_{1} \leftrightarrow R_{2}$$

Step-by-Step Solution

Answer

The matrix after swapping rows is \( \begin{bmatrix} 1 & 2 & 5 \\ 2 & -3 & -4 \end{bmatrix} \).

Step 1: Understand the Matrix

Identify the rows of the given augmented matrix. We have two rows, \( R_1 = \begin{bmatrix} 2 & -3 & -4 \end{bmatrix} \) and \( R_2 = \begin{bmatrix} 1 & 2 & 5 \end{bmatrix} \). The matrix currently looks like this: \[\begin{bmatrix}2 & -3 & -4 \1 & 2 & 5\end{bmatrix}\]

Step 2: Swap Rows

According to the problem, we need to perform the operation \( R_{1} \leftrightarrow R_{2} \), which means we will swap \( R_1 \) with \( R_2 \). After swapping, the new arrangement of rows is:\[\begin{bmatrix}1 & 2 & 5 \2 & -3 & -4\end{bmatrix}\]Now, \( R_1 \) is \( \begin{bmatrix} 1 & 2 & 5 \end{bmatrix} \) and \( R_2 \) is \( \begin{bmatrix} 2 & -3 & -4 \end{bmatrix} \).

Key Concepts

Augmented Matrix

An augmented matrix is a special type of matrix often used in linear algebra when dealing with systems of linear equations. It includes both the coefficients of the variables and the constants from the equations on the right side of the equal sign. The main purpose of using an augmented matrix is to organize the data systematically, making it easier to apply various mathematical operations to solve the system.
When looking at an augmented matrix, you'll typically see a vertical line dividing two parts. On the left side, you'll find the coefficients of the variables, and on the right, the constants. This setup allows you to efficiently perform operations like row swaps, row additions, and other types of manipulations that are vital for solving equations.
Using augmented matrices is efficient because they allow you to handle multiple equations as a single entity. For example, in the provided exercise, the matrix \(\begin{bmatrix} 2 & -3 & -4 \ 1 & 2 & 5 \end{bmatrix}\) represents two simultaneous equations. The numbers form a structured form that visualizes both the problem and the solution steps without writing out each equation separately.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces, linear transformations, and systems of linear equations. At its core, linear algebra deals with linear equations to allow for a deeper understanding of various mathematical systems and practical problems.
Linear algebra is crucial in numerous fields such as engineering, physics, computer science, and economics because it provides powerful tools for modeling and solving complex, real-world problems. This area of mathematics helps in understanding multi-dimensional spaces and finding optimized solutions through equations.

• Vectors and matrices are fundamental elements of linear algebra. While vectors can be thought of as points in space, matrices operate as tools to transform and manipulate these spaces effectively.
• Learning linear algebra offers you the ability to approach problems systematically, work with abstractions of real-world systems, and apply matrix operations to solve problems.

In the exercise presented, linear algebra principles are applied through matrix manipulation. By using the augmented matrix and practicing row operations, students explore foundational linear algebra concepts and solve system equations fluidly.

Elementary Row Operations

Elementary row operations are basic manipulations that can be applied to the rows of a matrix to simplify it or to find its solution. These operations include:

• Row swapping: Changing the position of two rows in the matrix. This does not influence the equations themselves but can help in the arrangement for further operations.
• Row multiplication: Multiplying each element of a row by a non-zero number, affecting both coefficients and constants proportionally.
• Row addition: Adding or subtracting the elements of one row to/from another, which is particularly useful for eliminating coefficients in systems of equations.

These operations are vital because they form the basis of methods like Gaussian elimination and Gauss-Jordan elimination, which solve systems of linear equations by transforming matrices into simpler forms, often triangular or reduced row-echelon form.
In the given exercise, the elementary row operation of swapping is used. It involves changing the order of rows, making it easier to proceed with other operations or interpretations needed to reach a solution. Understanding how to manipulate matrices through elementary row operations is key to effectively solving matrix-related problems in linear algebra.