Give an example of an (augmented) matrix **in row reduced echelon form** (RREF), that represents a linear system of equations that is **inconsistent**, and does **not have a pivot in every column.** 4 by 4

To construct an augmented matrix in row-reduced echelon form (RREF) that represents a linear system of equations which is inconsistent and does not have a pivot in every column, we need to satisfy two key conditions:

1. **Inconsistency:** This happens when there is a row in the augmented matrix where the left side (corresponding to the variables) consists entirely of zeros, but the right side (constant) is non-zero, i.e., \([0\ 0\ 0\ 0\ | \text{non-zero}]\).
  
2. **Not having a pivot in every column:** This means that at least one of the columns corresponding to the variables does not contain a leading 1 (a pivot).

### Example of such a matrix:

\[
\begin{pmatrix}
1 & 0 & 0 & 2 & | & 3 \\
0 & 1 & 0 & -1 & | & 2 \\
0 & 0 & 0 & 0 & | & 1 \\
0 & 0 & 0 & 0 & | & 0 \\
\end{pmatrix}
\]

### Explanation:

- **Row 1:** The first row has a pivot in the first column.
- **Row 2:** The second row has a pivot in the second column.
- **Row 3:** This row is inconsistent because it reads \(0x_1 + 0x_2 + 0x_3 + 0x_4 = 1\), which is impossible to satisfy, making the system inconsistent.
- **Row 4:** This row is entirely zero and thus doesn't affect the consistency.

**No Pivot in Every Column:**

- **Third Column (corresponding to \(x_3\)):** There is no pivot, as the third column consists of zeros in all rows.
  
This matrix meets the requirements of being in RREF, representing an inconsistent system, and not having a pivot in every column. 

Would you like to dive deeper into any aspect of this example?

### Related Questions:
1. What does the absence of a pivot in a column imply about the solution set of a linear system?
2. How does the presence of a row like \([0\ 0\ 0\ 0\ | \text{non-zero}]\) ensure inconsistency?
3. What are the steps to convert a general matrix to its row reduced echelon form?
4. How does one identify free variables from a matrix in RREF?
5. How does inconsistency manifest in terms of the original system of equations?

**Tip:** When analyzing a matrix in RREF, pay close attention to rows that indicate inconsistency, as these are key to understanding the solvability of the system.

Explore an example of an augmented matrix in row reduced echelon form (RREF) that represents a linear system of equations which is inconsistent and does not have a pivot in every column. Understand the conditions for inconsistency and the absence of pivots in matrices. Suitable for advanced college-level students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation