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:

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.