Use elementary row operations to write the augment coefficient matrix in reduced row echelon form: [(1,1,1,0),(1,1,15,0)]

We are given the augmented coefficient matrix:

\[
\begin{bmatrix}
1 & 1 & 1 & 0 \\
1 & 1 & 15 & 0
\end{bmatrix}
\]

Our goal is to use elementary row operations to transform this matrix into **reduced row echelon form (RREF)**.

### Step 1: Subtract row 1 from row 2 to eliminate the first element in row 2.
\[
R_2 = R_2 - R_1
\]

This gives:
\[
\begin{bmatrix}
1 & 1 & 1 & 0 \\
0 & 0 & 14 & 0
\end{bmatrix}
\]

### Step 2: Scale row 2 to make the leading coefficient (third entry) a 1.
\[
R_2 = \frac{1}{14}R_2
\]

This gives:
\[
\begin{bmatrix}
1 & 1 & 1 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\]

### Step 3: Eliminate the third element in row 1 by subtracting row 2 from row 1.
\[
R_1 = R_1 - R_2
\]

This gives:
\[
\begin{bmatrix}
1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\]

### Step 4: Further simplify by subtracting the second row from row 1 to get the final form.
Since the matrix already satisfies the conditions for reduced row echelon form (RREF), this is the final result:

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

This is the matrix in reduced row echelon form.

Would you like further details or explanations? Here are 5 questions related to this:

1. What is the difference between row echelon form and reduced row echelon form?
2. How do you determine when a matrix is in reduced row echelon form?
3. What are elementary row operations, and how are they used in solving systems of equations?
4. Can the reduced row echelon form help in finding the solution to a system of equations? If so, how?
5. Why is it important to scale rows when performing row operations?

**Tip:** Always ensure that leading entries are 1 in RREF and zeros in all other entries of their columns.

Learn how to transform the augmented matrix [(1,1,1,0),(1,1,15,0)] into reduced row echelon form (RREF) using elementary row operations. This solution covers key concepts in linear algebra, such as matrix operations and the RREF theorem. Suitable for college-level students studying linear algebra.

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