Math Problem Statement
Lose die aufgaben
Die aufgaben lösen
Solution
Let's go through each task step by step.
First task:
We are given a system of equations represented by the matrix:
-2 & 5 & 3 & 3 \\ -6 & 17 & 19 & 15 \\ 2 & -5 & -3 & -12 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} -4 \\ -2 \\ 19 \end{pmatrix}$$ We need to determine the number of solutions to this system. We will analyze the rank of the coefficient matrix and compare it with the augmented matrix to determine whether there are infinitely many solutions, exactly one solution, or no solution. The question options are: 1. Infinitely many solutions (2 free parameters) 2. Exactly three solutions (e.g., $$x_2 = 5$$) 3. Exactly one solution 4. Infinitely many solutions (1 free parameter) 5. No solution To solve, we can perform row reduction (Gaussian elimination) on the augmented matrix. I'll now proceed with this process. ### Second task: The second question presents the following system of equations: $$-6z - 8y = -60$$ $$-3z - y = -21$$ $$-8y = -24$$ We need to determine how many solutions this system has and find the values of $$x$$, $$y$$, and $$z$$. Let me go ahead and perform the calculations to determine the solution. numpy as np # Define the augmented matrix for the first system A1 = np.array([[-2, 5, 3, 3], [-6, 17, 19, 15], [2, -5, -3, -12]]) b1 = np.array([-4, -2, 19]) # Perform row reduction to check the rank of A1 from numpy.linalg import matrix_rank rank_A1 = matrix_rank(A1) augmented_matrix_1 = np.column_stack((A1, b1)) rank_augmented_1 = matrix_rank(augmented_matrix_1) # Now solve the second system # System 2 A2 = np.array([[0, -8, -6], [0, -1, -3], [0, -8, 0]]) # Matrix without x, y, z b2 = np.array([-60, -21, -24]) rank_A2 = matrix_rank(A2) augmented_matrix_2 = np.column_stack((A2, b2)) rank_augmented_2 = matrix_rank(augmented_matrix_2) rank_A1, rank_augmented_1, rank_A2, rank_augmented_2Here are the results of the analysis: ### First task (Matrix system): - The rank of the coefficient matrix is 3. - The rank of the augmented matrix is also 3. Since both ranks are equal and the rank is less than the number of variables (4 variables), this system has **infinitely many solutions** with **2 free parameters**. So, the correct answer is: **"unendlich viele Lösungen (2 freie Parameter)"**. ### Second task (System of equations): - The rank of the coefficient matrix is 2. - The rank of the augmented matrix is also 2. Since the rank of the augmented matrix equals the rank of the coefficient matrix and is less than the number of variables (3 variables), the system has **infinitely many solutions**. The correct answer is: **"inf" (infinitely many solutions)**. Would you like me to go into more detail or help with any further questions? Here are some related questions to expand your understanding: 1. How do you perform Gaussian elimination on augmented matrices? 2. How does the rank of a matrix relate to the number of solutions in a linear system? 3. What happens when the rank of the augmented matrix is greater than the rank of the coefficient matrix? 4. How can you find the general solution for systems with infinitely many solutions? 5. What is the geometric interpretation of having more variables than the rank of the matrix? **Tip:** When solving systems with infinitely many solutions, it's important to express the solutions in terms of free parameters to describe the entire solution space.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems
Matrix Rank
Gaussian Elimination
Formulas
Rank of Matrix
Gaussian Elimination
Theorems
Rank Theorem
Suitable Grade Level
Grades 10-12, College Level
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