Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. When using Gaussian elimination to solve a system of linear equations, you may conclude that the system is inconsistent before you complete the process of rewriting the augmented matrix in row-echelon form.

Step-by-Step Solution

Answer

The statement is true. During the Gaussian elimination process, the system may exhibit a characteristic row (0 … 0 | 1) that indicates inconsistency before the matrix is fully transformed into row-echelon form.

Step 1: Concept of Gaussian elimination

Gaussian elimination is a procedure to solve a system of linear equations. It includes three main steps: (1) Swap rows, (2) Multiply a row by a non-zero constant, \(3\) Add/subtract rows to each others, to eventually transform augmented matrix to row-echelon form.

Step 2: Concept of Inconsistent Systems

A system of linear equations is said to be inconsistent if it does not have a solution. For a system of linear equations represented in augmented matrix form through Gaussian elimination, an inconsistency would be revealed by a row of the form [0 ... 0 | 1], because it represents an equation like 0 = 1, which is impossible.

Step 3: Evaluate the Statement

Given the definition of inconsistency and the operation of Gaussian elimination, it's possible that, during execution of Gaussian elimination, we encounter a row of the form [0 ... 0 | 1]. So, even before completing the transformation to row-echelon form, we can already conclude that the system is inconsistent.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that share a common set of variables. To illustrate, imagine we are working with two equations representing lines in a two-dimensional space. The point where these lines intersect is the solution to this system, meaning the values of the variables satisfy both equations simultaneously. However, not every system has a clear point of intersection, and this is where solving becomes interesting. We use methods like Gaussian elimination to systematically find these points, if they exist, by manipulating the equations until they are in a form that is much simpler to interpret. For students, it is crucial to understand that the process of eliminating variables and isolating others is fundamental in solving these equations efficiently and effectively.

Gaussian elimination shines the spotlight on the matrix representation of a system. This compact arrangement organizes the coefficients and constants into a grid, known as an augmented matrix, setting the stage for the sequential transformations needed to get to a solution — or, in some cases, to realize that a solution does not exist.

Row-Echelon Form

When it comes to Gaussian elimination, the 'row-echelon form' is a term that denotes a specific state of the augmented matrix. An augmented matrix is in row-echelon form when it satisfies these conditions:

• Any zero rows are at the bottom of the matrix,
• The leading entry (first non-zero number from the left) of a row is always strictly to the right of the leading entry of the row above it,
• All entries directly below a leading entry are zeros.

Getting to this point helps clarify which variables can be solved for directly and which ones need further manipulation. Understanding this format is vital for students not just for solving systems of equations, but also for grasping the foundation of more complex linear algebra concepts. By aiming to bring an augmented matrix to this form, we often simplify the system to a point where the solutions become apparent through a process known as back-substitution. However, achieving row-echelon form is not always necessary for determining the consistency of the system, which leads to the subject of inconsistent systems.

Inconsistent Systems

In the world of linear equations, not all systems play nicely. An inconsistent system is one that does not have a solution. This happens, for instance, when two lines are parallel; no matter how long they extend, they will never meet — there is no point in common. When using Gaussian elimination, we may reveal this inconsistency long before we fully transform our matrix. For example, if we manipulate the matrix to a stage where we have a row of [0 ... 0 | 1], we've hit a mathematical dead end: it suggests that 0 equals 1, which is a clear contradiction.

The moment we encounter this row, we can stop our elimination process, because we've just learned something vital about our system — it's not solvable. Students frequently encounter inconsistency in more complex systems, so recognizing this early on can save substantial time and effort. Furthermore, this early detection of an inconsistent system underscores the efficiency of Gaussian elimination, exemplifying how it not only leads to solutions but also to conclusive evidence when no solutions exist.