Math Problem Statement

Is the following statement True or False? The linear system of equations represented by the matrix [[2, 1, 0, 1], [0, 0, 3, 8], [0, 4, 0, 2], [0, 0, 0, 0]] has infinitely many solutions.

Solution

The image shows a linear system of equations represented by a matrix. To determine if this system has infinitely many solutions, we need to analyze its structure by considering the rank of the coefficient matrix and the number of equations.

Here’s the matrix:

2 & 1 & 0 & 1 \\ 0 & 0 & 3 & 8 \\ 0 & 4 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ ### Analysis: 1. The system has 4 rows and 3 columns in the coefficient matrix (not counting the right-hand side, which represents the constants). 2. The last row is entirely zero, which indicates the presence of a dependent equation, suggesting the system might not be fully independent. 3. The rank of the coefficient matrix (number of non-zero rows in row-reduced form) appears to be 3, which is less than the number of variables (4 variables). ### Solution Characteristics: - Since the rank is less than the number of variables, this is an **underdetermined system**. - Underdetermined systems typically have **infinitely many solutions** because there are free variables. ### Conclusion: The statement "The linear system of equations has infinitely many solutions" is **True**. --- Would you like further details or have any questions? Here are five related questions to deepen your understanding: 1. What is the rank of a matrix, and how does it affect the number of solutions in a system of equations? 2. What is the difference between consistent, inconsistent, and dependent systems in linear algebra? 3. How can Gaussian elimination help to determine whether a system has infinitely many solutions? 4. How does the presence of free variables lead to infinitely many solutions in a system of equations? 5. Can a system have no solution even if there are zero rows in the matrix? **Tip:** When analyzing a matrix, always check for zero rows and compare the number of equations to the number of variables to predict the number of solutions.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Equations
Rank of Matrix
Solution of Linear Systems

Formulas

Rank of a Matrix
Determining free variables in linear systems

Theorems

Rank-Nullity Theorem
Consistent and Inconsistent Systems
Homogeneous and Non-Homogeneous Systems

Suitable Grade Level

Grades 10-12

Related Recommendation

Find m for a Linear System to Have Infinite Solutions

Understanding Systems of Equations with Row Reduction and Infinite Solutions

Solving Systems of Linear Equations: No Solution, One Solution, or Infinite Solutions

Understanding Systems of Linear Equations and Infinite Solutions

Solving Systems of Linear Equations with a Unique, No, or Infinitely Many Solutions