Problem 16 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

Row 2 has been subtracted by twice of Row 1, and Row 3 has been added by Row 1 in the original matrix. These are the row operations used to obtain the new row-equivalent matrix.

Step 1: Identify Changes between Matrices

Before identifying operations, let's first compare the original matrix and the new one. It's clear that the first row remains unchanged. However, the second and third rows are different. The original second row [2, -5, 1, -7] is now [0, -9, 7, -11] and the original third row [5, 4, -7, 6] is now [0, -6, 8, -4].

Step 2: Determine Operations for Row 2

Now think about what operation can transform the original second row into the new one. Since the first element has been changed from 2 to 0, evidently, Row 2 has been subtracted by twice Row 1 in the original matrix. Therefore, the operation for Row 2 is \( Row_2 New = Row_2 Old -2 \cdot Row_1 \).

Step 3: Determine Operations for Row 3

Similarly, to transform the original third row into the new one, since the first element also changes from 5 to 0, Row 3 has definitely been added by the original Row 1. Therefore, the operation for Row 3 is \( Row_3 New = Row_3 Old + Row_1 \).

Key Concepts

Elementary Row Operations

Understanding elementary row operations is crucial in matrix algebra and linear algebra. These operations are basic maneuvers you can perform on rows of a matrix to alter its contents while maintaining key properties. There are three types of elementary row operations: row switching, row multiplication, and row addition or subtraction (which includes the exercise we're discussing).

In the provided exercise, two row operations were used to transform the original matrix into a row-equivalent one. The exercise reveals how adding or subtracting multiples of one row to another can drastically change the matrix's appearance. But why do this? These operations help to simplify matrices, leading to easier methods for finding solutions to systems of linear equations, which are represented by the matrix.

To improve understanding, let's have a closer look at the operations mentioned in the solution:

Matrix Algebra

Row Operations in Practice

In matrix algebra, we manipulate matrices using defined operations to solve systems of equations or to find other properties like determinants and inverses. Remember that matrices represent linear transformations and systems, and manipulating them is a structured approach that underlies many practical applications, from computer graphics to engineering problems.

Solving Systems

Elementary row operations are especially useful in solving systems of equations. The exercise shows a step towards a row echelon form, which is a staging post towards finding a unique solution, if it exists. The 'unchanged' first row, along with the modified second and third rows, are steps towards this form. This method, as part of matrix algebra, can also lead to more advanced methods like Gaussian elimination or finding a matrix's rank.

Linear Algebra

Linear algebra is the area of mathematics that deals with vectors, vector spaces, and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

The row-equivalent matrices from the exercise demonstrate an important concept in linear algebra: operations on a matrix can alter its form without changing its solution space or the linear system it represents. This is why such matrices are 'equivalent.' They serve different visual or computational purposes while mathematically describing the same spatial relationships. It's like using different words to say the same thing.

In applications, linear algebra is the foundation for scientific computing and functional analysis, playing a significant role in machine learning, statistics, and physics. Getting comfortable with these operations opens doors to understanding more complex concepts and their utilization in various fields.