Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Can a \(2 \times 4\) augmented matrix whose entries are all nonzero real numbers represent an independent system of linear equations? Explain.

Step-by-Step Solution

Answer

No, a \(2 \times 4\) augmented matrix with non-zero real numbers cannot represent an independent system of linear equations because there are more variables than equations.

Step 1: Understand the relationship between equations and variables

In an independent system of linear equations, there must be at least as many equations as there are variables. This is because each equation provides information about the variables, helping to narrow down their possible values. If there are more variables than equations, then there will be multiple valid solutions for the variables, which can make the system dependent or inconsistent.

Step 2: Apply the concept to the given matrix

In this case, there are only 2 equations (corresponding to the 2 rows of the matrix), but 4 variables (corresponding to the 4 columns of the matrix). Because there are more variables than equations, this system cannot be independent.

Key Concepts

Augmented Matrix

An augmented matrix is a fundamental tool used to solve systems of linear equations. When setting up an augmented matrix, you are essentially merging the coefficients of the variables and the constants from the equations into a single matrix form. This format makes solving equations easier, especially when you use methods like row reduction.

In our given problem, the augmented matrix is of size \(2 \times 4\). This means it has 2 rows and 4 columns. The rows represent the equations, while the columns, apart from the last one, represent the variables in the system.

• The first 3 columns typically contain the coefficients of the variables.
• The last column represents the constants from the right side of the equations.

Understanding the structure of an augmented matrix helps in the visualization and manipulation needed to solve or analyze a system of equations.

System of Linear Equations

A system of linear equations is simply a set of equations with multiple variables. These systems can have a few different characteristics. They can be consistent or inconsistent, and dependent or independent.

• A consistent system means that there is at least one set of variable values that satisfies all the equations.
• An inconsistent system means that no such set of values exists.
• A dependent system usually has infinitely many solutions, as the equations are not uniquely identifying a single solution.
• An independent system has a unique solution, meaning just one set of values for the variables is possible.

In the context of our augmented matrix problem, there are 4 variables but only 2 equations, which results in a dependent system because multiple solutions can satisfy the couple of equations provided.

Variables and Equations Relationship

The relationship between variables and equations in a linear system is crucial to determine if the system is independent or dependent. Normally, to ensure an independent system, the number of equations should be equal to the number of variables or more.

If you have fewer equations compared to variables, as in our exercise with 2 equations and 4 variables, the system is called underdetermined. This happens because there isn't enough information to pinpoint the values of all variables uniquely.

• When the number of equations equals the number of variables, you have a chance for a unique solution resulting in an independent and usually a consistent system.
• When there are more variables than equations, multiple variable combinations can satisfy the equations, hence the system is dependent.

Therefore, understanding this relationship helps in determining the nature and the possible solutions of the system.