Math Problem Statement
Suppose n is a positive integer and a1, a2, . . . , an are real numbers. What condition must be satisfied for the linear system x1 − x2 = a1 x2 − x3 = a2 x3 − x4 = a3 ... xn−1 − xn = an−1 xn − x1 = an to be consistent? Write down the augmented matrix of this linear system. Assuming the system is consistent, find a general formula for its solutions.
Solution
Linear System Consistency Condition
We are given a linear system of equations involving real numbers and variables . The system is:
x_1 - x_2 &= a_1 \\ x_2 - x_3 &= a_2 \\ x_3 - x_4 &= a_3 \\ & \ \vdots \\ x_{n-1} - x_n &= a_{n-1} \\ x_n - x_1 &= a_n \end{aligned}$$ ### Augmented Matrix The augmented matrix of this system can be written by representing each equation as a row. Each row corresponds to the coefficients of the variables in the equations. Let's first write down the coefficient matrix of the system: $$\begin{pmatrix} 1 & -1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & -1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & -1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ -1 & 0 & 0 & 0 & \cdots & 1 \end{pmatrix}$$ The augmented matrix will be: $$\begin{pmatrix} 1 & -1 & 0 & 0 & \cdots & 0 & | & a_1 \\ 0 & 1 & -1 & 0 & \cdots & 0 & | & a_2 \\ 0 & 0 & 1 & -1 & \cdots & 0 & | & a_3 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & | & \vdots \\ -1 & 0 & 0 & 0 & \cdots & 1 & | & a_n \end{pmatrix}$$ ### Consistency Condition To determine the condition for consistency, let's consider the sum of all the equations: $$(x_1 - x_2) + (x_2 - x_3) + \dots + (x_n - x_1) = a_1 + a_2 + \dots + a_n$$ The left-hand side simplifies to: $$x_1 - x_2 + x_2 - x_3 + \dots + x_n - x_1 = 0$$ Thus, we have the condition: $$0 = a_1 + a_2 + \dots + a_n$$ Therefore, the system is consistent if and only if: $$\boxed{a_1 + a_2 + \dots + a_n = 0}$$ ### General Solution Assuming the system is consistent, we can now solve for the general solution. From the first equation: $$x_1 = x_2 + a_1$$ Substitute this into the second equation: $$x_2 = x_3 + a_2$$ Thus: $$x_1 = x_3 + a_1 + a_2$$ Repeating this process for each equation, we get a general formula for $$x_1$$: $$x_1 = x_n + a_1 + a_2 + \dots + a_{n-1}$$ Since the last equation is $$x_n - x_1 = a_n$$, we substitute the expression for $$x_1$$ to get: $$x_n - (x_n + a_1 + a_2 + \dots + a_{n-1}) = a_n$$ This simplifies to: $$-(a_1 + a_2 + \dots + a_n) = a_n$$ By the consistency condition, $$a_1 + a_2 + \dots + a_n = 0$$. Therefore, the general solution is that: $$x_1 = x_2 + a_1 = x_3 + a_1 + a_2 = \dots = x_n + (a_1 + a_2 + \dots + a_{n-1})$$ Thus, the solution has the form: $$x_i = c + \sum_{j=1}^{i-1} a_j \quad \text{for some constant } c \text{ and each } i = 1, 2, \dots, n.$$ ### Conclusion - **Consistency condition**: The system is consistent if $$a_1 + a_2 + \dots + a_n = 0$$. - **General solution**: The solutions are of the form $$x_i = c + \sum_{j=1}^{i-1} a_j$$, where $$c$$ is a constant. Would you like further clarification or details on any part of this solution? --- Here are some related questions for you to consider: 1. How can the condition $$a_1 + a_2 + \dots + a_n = 0$$ be interpreted geometrically? 2. What happens if $$a_1 + a_2 + \dots + a_n \neq 0$$? How would you modify the system? 3. How can we solve the system if the values of $$a_1, a_2, \dots, a_n$$ are explicitly given? 4. What are the implications of this system in terms of linear algebra, specifically in relation to null spaces? 5. How would this system change if we had a different number of variables? **Tip**: When analyzing a system of equations, always check for consistency by considering the sum or combination of all equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrices
Consistency of Equations
Formulas
Augmented Matrix of Linear System
Consistency Condition: a1 + a2 + ... + an = 0
General Solution Formula: xi = c + Σ(a_j)
Theorems
System of Equations Consistency Theorem
Suitable Grade Level
Undergraduate level (Linear Algebra course)
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