Let [[1,2,3,5], [2, 4, 8 ,12 ], [3,6,7,13 ]] Then rref(A) =?

Let's find the **Reduced Row Echelon Form (RREF)** of the matrix:

\[
A = \begin{bmatrix}
1 & 2 & 3 & 5 \\
2 & 4 & 8 & 12 \\
3 & 6 & 7 & 13
\end{bmatrix}
\]

### Steps to compute the RREF:

1. **Make the leading entry of the first row 1** (it's already 1, so we leave it unchanged).

2. **Use row operations to create zeros below the first column:**
    - Subtract 2 times row 1 from row 2:  
      \( R_2 \rightarrow R_2 - 2R_1 \)
    - Subtract 3 times row 1 from row 3:  
      \( R_3 \rightarrow R_3 - 3R_1 \)

   After performing these row operations:

   \[
   A = \begin{bmatrix}
   1 & 2 & 3 & 5 \\
   0 & 0 & 2 & 2 \\
   0 & 0 & -2 & -2
   \end{bmatrix}
   \]

3. **Make the leading entry in the second row (column 3) 1**:
    - Divide row 2 by 2:  
      \( R_2 \rightarrow \frac{1}{2} R_2 \)

   This gives:

   \[
   A = \begin{bmatrix}
   1 & 2 & 3 & 5 \\
   0 & 0 & 1 & 1 \\
   0 & 0 & -2 & -2
   \end{bmatrix}
   \]

4. **Use row operations to create zeros above and below the leading 1 in column 3:**
    - Add 2 times row 2 to row 3:  
      \( R_3 \rightarrow R_3 + 2R_2 \)
    - Subtract 3 times row 2 from row 1:  
      \( R_1 \rightarrow R_1 - 3R_2 \)

   After performing these operations, the matrix becomes:

   \[
   A = \begin{bmatrix}
   1 & 2 & 0 & 2 \\
   0 & 0 & 1 & 1 \\
   0 & 0 & 0 & 0
   \end{bmatrix}
   \]

5. **Final Check**: The matrix is now in reduced row echelon form (RREF).

\[
RREF(A) = \begin{bmatrix}
1 & 2 & 0 & 2 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

### Summary:
The RREF of the given matrix is:

\[
\begin{bmatrix}
1 & 2 & 0 & 2 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

---

Would you like more details on any of these steps or explanations about row operations? 

Here are a few related questions:
1. What does the RREF of a matrix represent geometrically?
2. How can I find the rank of a matrix using RREF?
3. What are the key differences between row echelon form (REF) and RREF?
4. Can all matrices be converted to RREF?
5. How can I use the RREF of a matrix to solve a system of linear equations?

**Tip**: RREF is particularly useful in solving systems of linear equations as it reveals the solutions directly when in the form of an augmented matrix!

Learn how to find the Reduced Row Echelon Form (RREF) of a matrix with a detailed step-by-step solution. This problem focuses on the concepts of linear algebra, row operations, and RREF, suitable for students in Grades 10-12.

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