Problem 38 - Precalculus with Limits

Question

In Exercises 35-38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Original Matrix \( \left[\begin{array}{rrrr} -1 & & -2 & & 3 & & -2 \\ 2 & & -5 & & 1 & & -7 \\ 5 & & 4 & & -7 & & 6 \\ \end{array}\right] \) New Row-Equivalent Matrix \( \left[\begin{array}{rrrr} -1 & & -2 & & 3 & & -2 \\ 0 & & -9 & & 7 & & -11 \\ 0 & & -6 & & 8 & & -4 \\ \end{array}\right] \)

Step-by-Step Solution

Answer

To change from the original matrix to the new row-equivalent matrix, two row operations were performed: the second row was replaced with (second row - 2 * first row) and the third row was replaced with (third row - 5 * first row).

Step 1: Identify the Change in Second Row

In the second row of the new matrix, the numbers are not same as in the original matrix. Specifically, 2 became 0, -5 became -9, 1 became 7, and -7 became -11. Looking at these changes, it's clear that the elementary row operation used here is that the second row was replaced with the result of (second row - 2 * first row). For instance, if we subtract -1*-2 from 2 we get 0 similarly, if we subtract -2*-2 from -5 we get -9 and similarly the rest.

Step 2: Identify the Change in Third Row

In the third row of the new matrix, the numbers are also not same as in the original matrix. Specifically, 5 became 0, 4 became -6, -7 became 8, and 6 became -4. Looking at these changes, it's also clear that the elementary row operation used here is that the third row was replaced with the result of (third row - 5 * first row). For instance, if we subtract -1*5 from 5 we get 0 similarly, if we subtract -2*5 from 4 we get -6 and similarly the rest.

Key Concepts

Row-Equivalent Matrices

Understanding row-equivalent matrices is a foundational concept in linear algebra that students encounter when first learning about matrix operations. Two matrices are considered row-equivalent if one can be transformed into the other using a series of elementary row operations. These operations include row switching, row multiplication (multiplying all entries in a row by a nonzero scalar), and row addition (adding or subtracting multiples of one row to another).

In the exercise provided, the transformation from the original matrix to the new row-equivalent matrix involved row addition operations. By performing specific row addition operations, defined by multiplying rows by a constant and then adding or subtracting them from other rows, we achieve a row-equivalent matrix that preserves the row space of the original matrix. These operations are crucial in simplifying matrices to solve systems of linear equations using methods like Gaussian elimination, which we shall explore in the next sections.

Matrix Algebra

Matrix algebra is a significant area of mathematics that deals with matrix operations and the rules that govern them. The elementary row operations used to create row-equivalent matrices are part of matrix algebra. When applying these operations to a matrix, it's important to understand that each operation corresponds to multiplying the original matrix by an elementary matrix from the left.

For example, when we modify the second row in our exercise by subtracting twice the first row from it, we are effectively multiplying the original matrix by an elementary matrix designed to perform this operation. These transformations are linear and maintain the structure of the solution set of the linear system associated with the matrices. By mastering matrix algebra, students gain a powerful tool for solving complex linear equations and for further studies in fields such as computer science, physics, and engineering.

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations. It is named after Carl Friedrich Gauss, a prolific German mathematician. This algorithmic approach transforms a matrix into row-echelon form using elementary row operations. From there, one can proceed to reduced row-echelon form, which makes it possible to read off the solutions or to determine if the system is inconsistent or has infinitely many solutions.

In the context of our exercise, the elementary row operations performed to achieve the new matrix can be seen as steps in the process of Gaussian elimination. The ultimate goal is to reach a point where back substitution can be used to find the values of the unknowns. Students who understand Gaussian elimination can apply it not only to solve linear systems but also to find matrix inverses and determine rank, thereby unlocking a deeper understanding of linear algebra and its applications in various scientific and engineering problems.