Math Problem Statement
Solution
The problem asks to solve and discuss the linear system of equations provided over the real numbers field. Here's a breakdown of the system:
\text{Given the system:} \\ 1. & \; mx_1 + x_2 + x_3 + x_4 = 1 \\ 2. & \; x_1 + mx_2 + x_3 + x_4 = 1 \\ 3. & \; x_1 + x_2 + mx_3 + x_4 = 1 \\ 4. & \; x_1 + x_2 + x_3 + mx_4 = 1, \\ \text{where $$m$$ is a real parameter.} \end{aligned}$$ ### Steps: 1. **Matrix Representation:** The system can be represented in matrix form as: $$A \mathbf{x} = \mathbf{b},$$ where $$A = \begin{bmatrix} m & 1 & 1 & 1 \\ 1 & m & 1 & 1 \\ 1 & 1 & m & 1 \\ 1 & 1 & 1 & m \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}.$$ 2. **Determinant of $$A$$:** To determine the nature of the solutions, calculate $$\det(A)$$. If $$\det(A) \neq 0$$, the system has a unique solution. If $$\det(A) = 0$$, analyze further for dependencies or inconsistencies. The determinant of $$A$$ is: $$\det(A) = \begin{vmatrix} m & 1 & 1 & 1 \\ 1 & m & 1 & 1 \\ 1 & 1 & m & 1 \\ 1 & 1 & 1 & m \end{vmatrix}.$$ This is computed using the formula for determinants: $$\det(A) = m^4 - 4m^3 + 6m^2 - 4m.$$ Factorizing: $$\det(A) = m(m - 1)^3.$$ 3. **Discussion on $$m$$:** - If $$m \neq 0$$ and $$m \neq 1$$, $$\det(A) \neq 0$$, and the system has a unique solution. - If $$m = 0$$, $$\det(A) = 0$$, and the matrix $$A$$ becomes singular. Analyze rank and consistency for solution types. - If $$m = 1$$, $$\det(A) = 0$$, and the matrix $$A$$ also becomes singular. Further analyze the rank for solution types. Would you like a detailed computation of solutions for specific cases, or shall we verify the nature of the solutions? Here are 5 follow-up questions to deepen understanding: 1. What happens to the solutions when $$m = 0$$? 2. Why does $$m = 1$$ lead to a different type of solution than $$m \neq 1$$? 3. Can the system have infinitely many solutions for any value of $$m$$? 4. How does the rank of $$A$$ affect the solution when $$\det(A) = 0$$? 5. Would a similar system work for complex numbers instead of real numbers? **Tip:** Always calculate the determinant first to analyze the uniqueness of solutions in linear systems!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrix Rank
Formulas
Determinant calculation: \(\det(A) = m^4 - 4m^3 + 6m^2 - 4m\)
Matrix form of a system: \(A \mathbf{x} = \mathbf{b}\)
Theorems
Unique solution criterion: \(\det(A) \neq 0\)
Rank-nullity theorem for \(\det(A) = 0\) cases
Suitable Grade Level
Undergraduate Mathematics or Advanced High School
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