Math Problem Statement

Consider the following augmented matrix of a system of linear equations: [ 1 -1 -1 1 ] [ 0 1 1 1 ] [ 0 1 1 1 ] The system has: (a) infinitely many solutions with one free variable (b) infinitely many solutions with two free variables (c) unique solution (d) no solutions

Solution

We are given an augmented matrix of a system of linear equations:

1 & -1 & -1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \end{bmatrix}$$ To determine the nature of the solution, we will perform **row reduction** (Gaussian elimination) to transform this matrix into row echelon form. This will help us identify the number of free variables or whether the system has a unique solution or no solutions. Let's begin by working on the row reduction. Starting with the matrix: $$\begin{bmatrix} 1 & -1 & -1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \end{bmatrix}$$ Notice that rows 2 and 3 are identical. Subtracting Row 2 from Row 3: $$R_3 \rightarrow R_3 - R_2$$ Results in: $$\begin{bmatrix} 1 & -1 & -1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Now that the matrix is in **row echelon form**, let's analyze the system. - The third row is all zeros, indicating that one equation is redundant. - The first and second rows provide two independent equations with three unknowns, implying **one free variable**. Thus, the system has infinitely many solutions with **one free variable**. ### Conclusion: The correct answer is **(a) infinitely many solutions with one free variable**. --- Would you like further clarification or details on the row reduction process? Here are 5 related questions for deeper understanding: 1. How do we identify free variables in a system of equations? 2. What is the significance of a row of zeros in an augmented matrix? 3. How does Gaussian elimination help solve systems of equations? 4. Can a system with more unknowns than equations have a unique solution? 5. What are the possible solution types for a system of linear equations? **Tip:** Always check for row redundancy when solving systems of equations—it can simplify the problem by reducing the number of independent equations.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Gaussian Elimination
Augmented Matrix

Formulas

Row operations for Gaussian elimination
Echelon form

Theorems

Rank theorem
Solution types of a system of linear equations

Suitable Grade Level

Undergraduate Linear Algebra or advanced high school algebra

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