Math Problem Statement

how am i supposed to solve this? can you show me the solution?

How am I supposed to solve this? Can you show me the solution?

Solution

To solve this linear system represented by the augmented matrix, we need to analyze the matrix structure and use Gaussian elimination to determine the solution set.

Step-by-Step Solution

1. Matrix Analysis:

The matrix provided is:

1 & 1 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & | & m \\ 0 & 0 & 1 & 2 & 0 & 0 & \cdots & 0 & 0 & | & m - 1 \\ 0 & 0 & 0 & 0 & 1 & 3 & \cdots & 0 & 0 & | & m - 2 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & | & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 1 & m & | & 1 \\ \end{pmatrix}$$ where each row follows a distinct pattern with different positions filled with constants. #### 2. **Recognize Pivot Positions**: Each row has a leading entry in a different column, which indicates this matrix is in a form close to row echelon form. By inspecting each row: - The first row has a pivot in the first column. - The second row has a pivot in the third column. - The third row has a pivot in the fifth column. - This pattern continues, suggesting we might be dealing with a system where each equation has its own pivot column. #### 3. **Identify Free and Dependent Variables**: Given the structure, it appears that this matrix allows us to solve for variables iteratively, setting up each row to solve one variable in terms of previous ones. #### 4. **Back-Substitution**: Since this matrix resembles an upper triangular form, we can proceed by substituting values back from the bottom row upwards. #### 5. **Specific Solution and Parameter Dependency**: - Each variable $$x_i$$ will depend on the parameter $$m$$ and potentially any free variables, given the structure of each row's equation. - The solution set will vary based on the values of $$m$$ due to the dependency introduced by each constant in the rightmost column of the matrix. ### Example: Let's go through the first few rows to illustrate the solution process. Suppose we have: - Row 1: $$x_1 + x_2 = m$$, which gives $$x_1 = m - x_2$$. - Row 2: $$x_3 + 2x_4 = m - 1$$, so $$x_3 = m - 1 - 2x_4$$. - Row 3: $$x_5 + 3x_6 = m - 2$$, so $$x_5 = m - 2 - 3x_6$$. - And so on. Each variable can be expressed in terms of a free variable in this form. This indicates a parametric solution set where some variables depend on others (free variables), and the constants $$m, m-1, m-2, \ldots$$ define the solution space's specifics. ### Summary of Solution Process 1. Identify the pivot structure in each row. 2. Set up expressions for each pivot variable in terms of free variables. 3. Substitute back to express the dependent variables accordingly. 4. The solution set will vary based on the parameter $$m$$, defining different solution spaces for varying values of $$m$$. Would you like a further breakdown of each row, or do you have specific questions about the process? Here are some questions to deepen your understanding: 1. How do we determine free and pivot variables in a matrix? 2. How does the parameter $$m$$ affect the solution set? 3. What conditions on $$m$$ could make this system inconsistent? 4. How would Gaussian elimination proceed if there were zero rows? 5. How does a parametric solution differ from a single unique solution? **Tip**: When solving augmented matrices, focus on identifying pivots early, as this helps simplify the steps for Gaussian elimination or back-substitution.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Operations

Formulas

Gaussian Elimination
Back-Substitution

Theorems

Row Echelon Form
Existence and Uniqueness Theorem

Suitable Grade Level

Undergraduate Mathematics

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