Consistent and inconsistent matrix proving with example

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 \([A|B]\), where \(A\) is the coefficient matrix, and \(B\) 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:
     \[
     [0 \; 0 \; \cdots \; 0 \; | \; c] \quad \text{where} \; c \neq 0,
     \]
     the system is inconsistent (no solution exists).
   - Otherwise, the system is consistent.

---

### Example: **Consistent Matrix**
Consider the system of equations:
\[
x + 2y = 5 \quad \text{and} \quad 2x - y = 4
\]

The augmented matrix is:
\[
\begin{bmatrix}
1 & 2 & | & 5 \\
2 & -1 & | & 4
\end{bmatrix}
\]

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.

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### 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.

Understand how to determine if a matrix is consistent or inconsistent with step-by-step examples. This guide covers key concepts in linear algebra, including Gaussian elimination, row reduction, and matrix consistency rules. Suitable for high school and undergraduate students.

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