Problem 101 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It The augmented matrix represents a system of linear equations (in the variables \(x, y,\) and \(z\) ) that has been reduced using Gauss-Jordan elimination. Write a system of three equations in three variables with nonzero coefficients that is represented by the reduced matrix. (There are many correct answers.) $$\left[\begin{array}{llllr} 1 & 0 & 3 & \vdots & -2 \\ 0 & 1 & 4 & \vdots & 1 \\ 0 & 0 & 0 & \vdots & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The system of equations represented by the reduced augmented matrix is: \[x + 3z = -2\], \[y + 4z = 1\], \[0 = 0\]

Step 1: Interpret the First Row

The first row is given as [1, 0, 3, -2]. This can be translated into a linear equation as: \(x + 0y + 3z = -2\)

Step 2: Interpret the Second Row

The second row is given as [0, 1, 4, 1]. This can be translated into a linear equation as follows: \(0x + y + 4z = 1\)

Step 3: Interpret the Third Row

The third row is given as [0, 0, 0, 0]. This indicates that the equation has no variables and this can be translated to: \(0 = 0\), which is always true and effectively introduces no new information to the system of equations.

Key Concepts

System of Linear Equations

Understanding the fundamentals behind a system of linear equations is crucial for grasping more advanced concepts in linear algebra. Simply put, a system of linear equations is a collection of one or more linear equations involving the same set of variables. Take, for instance, equations that depict various lines on a graph; when we talk about a system of these, we're looking for points where the lines intersect. Such points present solutions that satisfy all the equations simultaneously.

The system can be classified based on the number of solutions it has: a unique solution if there's only one point of intersection, no solution if there are parallel lines with no point of intersection, or infinitely many solutions if the lines lie on top of each other. In our exercise, the presence of the third row as a zero row (an equation represented by 0 = 0) signifies that there's not going to be a unique solution to the system.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations, which is especially helpful when using a method called Gauss-Jordan elimination to find solutions. It consists of the coefficients of the variables and the constants from the equations bundled together into matrix form. The line or vertical bar traditionally separates the coefficient section of the matrix from the constants, reminiscent of how an equation separates variables from constant terms.

By the end of the Gauss-Jordan process, an augmented matrix should ideally have what's known as reduced row-echelon form, where the leading entry (first non-zero number from the left) in each non-zero row is a 1 (pivot), and all the entries above and below the pivots are zeros. In the exercise, the matrix represents a reduced system of equations, which we can interpret to find the solutions to the original system.

Linear Algebra

The pillar behind solving systems of equations lies in the realm of linear algebra, a significant branch of mathematics. Linear algebra focuses on vector spaces, linear mappings, and the resolution of linear systems. The techniques developed here are widespread in fields such as engineering, physics, computer science, and economics because they often deal with large amounts of data that need to be organized and solved efficiently.

Gauss-Jordan elimination is one such technique that streamlines the process of finding solutions to systems of linear equations. It involves a sequence of operations performed on the augmented matrix to simplify it to a form where the variables’ values become apparent or the nature of the solutions is easily discernible.

Variables in Linear Equations

In a linear equation, variables are the unknown quantities we aim to solve for. Variables are typically represented by letters like x, y, and z and are accompanied by numerical coefficients in an equation. The number of variables in linear equations corresponds to the dimensionality of the solution space – this is why in two-variable systems, we graph lines on a two-dimensional plane, whereas, with three variables, we'd be looking at a three-dimensional space.

Variables can also be independent or dependent; the Gauss-Jordan elimination helps us to identify this by transforming the system into a form where these relationships are clearer. In the given matrix, x and y are independent whereas z is a free variable; it's not fixed and can take an infinite number of values since it is associated with the zero row, leading to infinitely many solutions involving z.