Math Problem Statement
Consistent and inconsistent matrix proving with example
Solution
To determine whether a matrix is consistent or inconsistent, we examine whether a system of linear equations represented by the matrix has a solution. A matrix is considered consistent if the system has at least one solution; otherwise, it is inconsistent.
Steps to Prove Consistency or Inconsistency
- Form the Augmented Matrix: Represent the system of equations as an augmented matrix , where is the coefficient matrix, and is the column of constants.
- Row Reduce: Perform Gaussian elimination or row-reduction to bring the matrix to row echelon form or reduced row echelon form.
- Analyze Rows:
- If any row in the augmented matrix has the form: the system is inconsistent (no solution exists).
- Otherwise, the system is consistent.
Example: Consistent Matrix
Consider the system of equations:
The augmented matrix is:
1 & 2 & | & 5 \\ 2 & -1 & | & 4 \end{bmatrix}$$ Performing row reduction: 1. Subtract $$2$$ times the first row from the second: $$\begin{bmatrix} 1 & 2 & | & 5 \\ 0 & -5 & | & -6 \end{bmatrix}$$ 2. Simplify the second row: $$\begin{bmatrix} 1 & 2 & | & 5 \\ 0 & 1 & | & \frac{6}{5} \end{bmatrix}$$ This system has solutions ($$x$$ and $$y$$), hence **consistent**. --- ### Example: **Inconsistent Matrix** Consider the system of equations: $$x + 2y = 5 \quad \text{and} \quad 2x + 4y = 10$$ The augmented matrix is: $$\begin{bmatrix} 1 & 2 & | & 5 \\ 2 & 4 & | & 10 \end{bmatrix}$$ Performing row reduction: 1. Subtract $$2$$ times the first row from the second: $$\begin{bmatrix} 1 & 2 & | & 5 \\ 0 & 0 & | & 0 \end{bmatrix}$$ This row reduction shows no contradiction ($$0 = 0$$), hence the system is **consistent** (infinite solutions). Now, if we modify the second equation to: $$2x + 4y = 11$$ The augmented matrix becomes: $$\begin{bmatrix} 1 & 2 & | & 5 \\ 2 & 4 & | & 11 \end{bmatrix}$$ Performing row reduction: 1. Subtract $$2$$ times the first row from the second: $$\begin{bmatrix} 1 & 2 & | & 5 \\ 0 & 0 & | & 1 \end{bmatrix}$$ This leads to a contradiction ($$0 = 1$$), hence the system is **inconsistent**. --- ### Summary - **Consistent**: Solutions exist. - **Inconsistent**: No solutions exist. --- #### Do you want further examples or explanations? ### Related Questions: 1. What are the conditions for a matrix to have infinitely many solutions? 2. How do free variables play a role in consistent systems? 3. What is the difference between a homogeneous and non-homogeneous system? 4. How does the determinant of a matrix relate to system consistency? 5. How do we identify dependent and independent rows in a matrix? ### Tip: Always verify row-reduction calculations step-by-step to avoid introducing errors when determining consistency.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Consistency
Formulas
Gaussian Elimination
Row Echelon Form
Reduced Row Echelon Form
Theorems
Consistency Theorem for Linear Systems
Row Reduction Properties
Suitable Grade Level
Grades 11-12, Undergraduate