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1. Given the following matrix, is it in row reduced form 

Let's analyze the augmented matrix to check if it is in row reduced form:

\[
\left( \begin{array}{cccc|c}
1 & 1 & 0 & 0 & -5 \\
0 & 0 & 1 & 0 & 3 \\
\end{array} \right)
\]

**Row reduced form (RREF) criteria**:
1. **Leading entry in each non-zero row is 1 (pivot)**.
2. **Each leading 1 is the only non-zero entry in its column**.
3. **Leading 1s appear to the right as you move down the rows**.
4. **Rows consisting of all zeroes are at the bottom (not applicable here)**.

### Check the given matrix:
1. The first row has a leading 1 in the first column, which is fine.
2. The second row has a leading 1 in the third column.
3. The entries above and below these leading 1s are all zeros, except for the second column of the first row, which contains a non-zero value (1). 

**Conclusion**:
This matrix is not in row-reduced form (RREF) because there is a non-zero entry in the second column of the first row where there should only be zeroes (according to the second rule of RREF).

Would you like me to show how to transform this matrix into row-reduced form?

### Follow-up Questions:
1. What are the steps to convert this matrix to row-reduced form?
2. How does row-echelon form differ from row-reduced form?
3. Can a matrix have multiple row-reduced forms?
4. How does solving a system of equations relate to matrix row reduction?
5. What happens if we apply a row swap in this matrix?

### Tip:
When reducing matrices, always aim to get leading 1s and ensure no other non-zero values exist in their columns!

Given the following matrix, is it in row reduced form: (1 1 0 -5; 0 0 1 3)?

Analyze a matrix to check if it is in row-reduced echelon form (RREF). Understand the criteria for RREF, including leading 1s, zeroes in specific positions, and row ordering. Learn how this matrix does not meet the criteria and how to correct it.

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