Math Problem Statement
Solve it
Solution
from sympy import symbols, Eq, solve, Matrix
Define symbols
x, y, z, lam, mu = symbols('x y z lam mu')
Define the equations in augmented matrix form
A = Matrix([ [1, 2, 1, 8], [2, 1, 3, 13], [-3, -4, -2, mu] ])
Coefficient matrix
coeff_matrix = A[:, :-1]
Check determinant of the coefficient matrix
det_coeff_matrix = coeff_matrix.det()
Compute rank of the coefficient matrix and augmented matrix
rank_coeff_matrix = coeff_matrix.rank() rank_augmented_matrix = A.rank()
(det_coeff_matrix, rank_coeff_matrix, rank_augmented_matrix)(-5, 3, 3)(-5, 3, 3)successThe determinant of the coefficient matrix is , and the ranks of both the coefficient matrix and the augmented matrix are equal to 3. This implies:
- The system has a unique solution if does not affect the rank condition.
- For specific values of and , the rank conditions might differ, leading to no solution or infinite solutions.
To fully solve for the conditions where (i) no solution, (ii) a unique solution, and (iii) an infinite number of solutions arise, I will further analyze specific conditions on and .# Substitute lambda and check the system for different conditions
Investigating for unique solution condition first
det_results = solve(det_coeff_matrix, lam) infinite_solutions_condition = Eq(rank_coeff_matrix, rank_augmented_matrix) no_solution_condition = rank_coeff_matrix < rank_augmented_matrix
Solve conditions for lambda and mu for unique, no, and infinite solutions
unique_condition = (rank_coeff_matrix == rank_augmented_matrix) and det_coeff_matrix!= Cell In[2], line 8 unique_condition = (rank_coeff_matrix == rank_augmented_matrix) and det_coeff_matrix!= ^ SyntaxError: invalid syntax
failed_with_in_kernel_exceptionI encountered a syntax issue in defining the conditions for unique, infinite, and no solutions. Let me correct this and continue solving the problem to provide clear results.# Check determinant for when unique solutions occur (non-zero determinant) unique_solution_condition = det_coeff_matrix != 0
Infinite solutions occur when rank of the coefficient matrix = rank of augmented matrix < number of variables
infinite_solution_condition = rank_coeff_matrix == rank_augmented_matrix and rank_coeff_matrix < 3
No solution occurs when rank of coefficient matrix < rank of augmented matrix
no_solution_condition = rank_coeff_matrix < rank_augmented_matrix
unique_solution_condition, infinite_solution_condition, no_solution_condition(True, False, False)(True, False, False)successThe results indicate the following:
-
Unique solution condition: The system has a unique solution as long as the determinant of the coefficient matrix is non-zero, which holds in this case ().
-
Infinite solutions condition: There are no infinite solutions because the rank of the coefficient matrix equals the number of variables (3).
-
No solution condition: There is no inconsistency since the rank of the coefficient matrix equals the rank of the augmented matrix.
Thus, the system has a unique solution for all values of and under these conditions. Let me know if you want further breakdown or additional questions.
Questions for exploration:
- How do changes in affect the solutions?
- What if or are set to specific values?
- Can we explicitly find the solutions for ?
- How is the determinant linked to system consistency?
- Can we illustrate this solution graphically?
Tip:
Always check the determinant and rank conditions when analyzing linear systems for consistency and the number of solutions.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Determinants
Matrix Rank
Formulas
Determinant of a matrix: det(A)
Rank of a matrix: rank(A)
Conditions for unique solutions: det(A) ≠ 0
Conditions for infinite solutions: rank(Coefficient Matrix) = rank(Augmented Matrix) < number of variables
Conditions for no solution: rank(Coefficient Matrix) < rank(Augmented Matrix)
Theorems
Cramer's Rule
Rank-Nullity Theorem
Rouché–Capelli Theorem
Suitable Grade Level
Undergraduate Level (Linear Algebra)
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