Problem 10 - Precalculus

Question

The following matrices are in reduced row echelon form. Determine the solution of the corresponding system of linear equations or state that the system is inconsistent. $$ \left[\begin{array}{llll|l} 1 & 0 & 0 & 3 & 0 \\ 0 & 1 & 2 & 6 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The system is inconsistent and has no solution.

Step 1: Understand the System of Equations

The augmented matrix represents a system of equations of the form \( \begin{aligned} x_1 + 3x_4 &= 0, \ x_2 + 2x_3 + 6x_4 &= 0, \ 0x_1 + 0x_2 + 0x_3 + 0x_4 &= 1. \end{aligned} \)

Step 2: Identify the Problematic Row

The third row, \(0 = 1\), is a contradiction. It implies that there are no solutions as no values for variables can satisfy this equation.

Step 3: Conclusion on Consistency

Given the contradiction in the third row, the system of equations is inconsistent and has no solution.

Key Concepts

Reduced Row Echelon Form

The Reduced Row Echelon Form (RREF) is a form of a matrix where certain conditions are met. Each leading entry (the first non-zero number from the left, in a row) must be 1. In any column containing a leading 1, all other entries of that column must be zero.
Additionally, each leading 1 is to the right of any leading 1s in previous rows, and rows of all zeroes are placed at the bottom of the matrix. This structure simplifies solving systems of linear equations. When a matrix is in reduced row echelon form, it clearly reflects the solutions of the corresponding linear system, or indicates if the system is inconsistent.

For instance, in order to solve an augmented matrix that is in reduced row echelon form, you simply observe the values on the right side of the augmented part; these directly give you the solutions if they are consistent, or reveal inconsistencies.

Inconsistent System

An inconsistent system of equations is one where no set of variable values can satisfy all the equations simultaneously.When you encounter an inconsistent system, it means there are no solutions that work for every equation in the system.
In terms of matrices, an inconsistent system often reveals itself when you have a row resembling \[0x_1 + 0x_2 + ... + 0x_n = c\] where \(c\) is a non-zero number, indicating an impossible situation.
Contradictory equations occur when the same variable terms lead to dissimilar outcomes, resulting in parallel but non-overlapping lines in a graphical representation.Such systems are critical to identify because, despite appearing solvable at first glance, they essentially terminate any search for solutions.

Augmented Matrix

An augmented matrix is an essential concept in linear algebra, as it's used to represent systems of linear equations.It combines the coefficients of the variables from each equation and their respective constant terms into one rectangular array.By placing these constants in the last column, you effectively streamline operations on the matrix.

For example, consider the system of equations:\[\begin{aligned}x + 3y &= b_1 \x - y &= b_2\end{aligned}\]This system can be written as the augmented matrix:\[\left[ \begin{array}{cc|c}1 & 3 & b_1 \1 & -1 & b_2\end{array} \right]\]Working with augmented matrices, especially in processes like row reduction to reduced row echelon form, makes it easier to analyze and solve systems, or identify whether they're inconsistent.

Contradiction in Linear Algebra

A contradiction in linear algebra happens when there is an assertion that is impossible to reconcile with itself.In terms of systems of linear equations, this typically surfaces as an equation such as \[0x_1 + 0x_2 + ... + 0x_n = 1\]This indicates that no combination of the variable values can satisfy this equation since no amount of multiplying and adding zero can result in one.

Such contradictions are crucial to recognize because they instantly reveal that a system of equations is inconsistent.These are among the most common reasons a linear system might not have a solution.Spotting a contradiction simplifies your task; it swiftly ends attempts to find solutions and points to the need to re-evaluate the system.