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

  1. Form the Augmented Matrix: Represent the system of equations as an augmented matrix [AB][A|B], where AA is the coefficient matrix, and BB is the column of constants.
  2. Row Reduce: Perform Gaussian elimination or row-reduction to bring the matrix to row echelon form or reduced row echelon form.
  3. Analyze Rows:
    • If any row in the augmented matrix has the form: [0  0    0    c]where  c0,[0 \; 0 \; \cdots \; 0 \; | \; c] \quad \text{where} \; c \neq 0, the system is inconsistent (no solution exists).
    • Otherwise, the system is consistent.

Example: Consistent Matrix

Consider the system of equations: x+2y=5and2xy=4x + 2y = 5 \quad \text{and} \quad 2x - y = 4

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.

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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