Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. \(\begin{array}{ll}\text { Original Matrix } & \text { New Row-Equivalent Matrix }\end{array}\) $$ \left[\begin{array}{rrr} 3 & -1 & -4 \\ -4 & 3 & 7 \end{array}\right] \quad\left[\begin{array}{rrr} 3 & -1 & -4 \\ 5 & 0 & -5 \end{array}\right] $$

Step-by-Step Solution

Answer

The elementary row operation applied is three times row one added to row two, depicted as '3R1 + R2 -> R2'.

Step 1: Analyze row changes between matrices

First, analyze the transformation from row 2 in the original matrix to row 2 in the new matrix. The original row 2 is \(-4, 3, 7\) and the new row 2 is \(5, 0, -5\). This suggests some form of addition or subtraction involving other rows.

Step 2: Identify the specific row operation

Row 1 (\(3, -1, -4\)) is constant in both matrices. Due to the change in row 2, it's clear that a certain multiple of row 1 is added to row 2. To figure out the specific multiple, look at the changes in the respective elements of row 2. The change from -4 to 5 involves adding 9, the change from 3 to 0 involves subtracting 3, and the change from 7 to -5 involves subtracting 12. These changes correspond to multiplying row 1 by 3 and adding to row 2: \(3 * R1 + R2 = new R2\). Hence, this seems to be the row operation applied.

Key Concepts

Row-Equivalent Matrices

Understanding row-equivalent matrices is a cornerstone in the study of linear algebra. These matrices may look different at first glance, but through a series of elementary row operations, one can be transformed into the other. In the given exercise, you're faced with recognizing how one matrix is converted into a row-equivalent counterpart.

To become row-equivalent, the original matrix has undergone specific modifications that maintain the essence of its row space. The row space is the set of all possible linear combinations of its rows, and for two matrices to be row-equivalent, they must have the same row space. In the textbook example, we deduced that row 2 of the original matrix was manipulated whilst row 1 remained unaltered. The transformation implied that a multiple of row 1 was added to row 2. This operation preserves the system of linear equations that the matrix represents, which is why they are considered row-equivalent.

Identifying row-equivalent matrices is crucial for numerous applications such as solving linear systems and understanding the rank of a matrix. Additionally, row operations help in achieving reduced row echelon form, an important step in solving systems of equations.

Matrix Transformation

When we talk about matrix transformation, we're referring to the methods by which we can systematically modify a matrix. These modifications can simplify a matrix or change its form to reveal more information about the system it represents. In the context of the problem from the textbook, matrix transformation involves applying elementary row operations to arrive at a row-equivalent matrix.

Elementary row operations are the backbone of matrix transformation and include operations such as row swapping, row multiplication (multiplying a row by a non-zero scalar), and row addition (adding multiples of one row to another row). In our exercise, the latter operation was used, where we took multiples of row 1 and added them to row 2, transforming the original matrix into its new form.

Such transformations are pivotal in many areas of linear algebra, including determining the inverse of a matrix and performing Gaussian elimination. It's essential to apply these operations correctly to ensure the matrices remain row-equivalent, maintaining their intrinsic properties while altering their outward appearance.

Linear Algebra

The field of linear algebra is vast and the backbone of many concepts in mathematics and applied sciences. It deals with vectors, vector spaces, linear mappings, and systems of linear equations. The elementary row operations we've discussed are practical tools within this discipline, providing a structured way to manipulate matrices that represent these linear systems.

Understanding linear algebra is fundamental to areas such as computer science, engineering, physics, and more. The matrix transformations we observe through elementary row operations help to simplify complex problems, making the systems easier to interpret and solve. In the textbook problem, applying a row operation to achieve a row-equivalent matrix is a prime example of linear algebra in action.

With each step taken in a matrix transformation, we're engaging with principles of linear algebra, from simple operations to more complex concepts like vector spaces and linear transformations. Mastery of these basics is essential for any student aiming to comprehend more intricate mathematical frameworks and their real-world applications. Linear algebra also forms the theoretical foundation for modern topics such as machine learning and data science, making it an indispensable part of mathematical education.