Math Problem Statement

Solve it

Investigate for what values of λ and μ the following system of linear equations will have (i) no solution, (ii) a unique solution, and (iii) an infinite number of solutions: 1. x + 2y + z = 8 2. 2x + y + 3z = 13 3. 3x + 4y - λz = μ

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 5-5, and the ranks of both the coefficient matrix and the augmented matrix are equal to 3. This implies:

  1. The system has a unique solution if μ\mu does not affect the rank condition.
  2. For specific values of λ\lambda and μ\mu, 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 λ\lambda and μ\mu.# 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:

  1. 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 (50-5 \neq 0).

  2. Infinite solutions condition: There are no infinite solutions because the rank of the coefficient matrix equals the number of variables (3).

  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 λ\lambda and μ\mu under these conditions. Let me know if you want further breakdown or additional questions.

Questions for exploration:

  1. How do changes in μ\mu affect the solutions?
  2. What if λ\lambda or μ\mu are set to specific values?
  3. Can we explicitly find the solutions for x,y,zx, y, z?
  4. How is the determinant linked to system consistency?
  5. 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)