perform elementary row operations, A:In = [1 1 1 : 1 0 0 0] [2 3 -1 : 0 1 0] [-4 7 4 : 0 0 1]

We are tasked with performing elementary row operations on the augmented matrix \( A : I_n \), where:

\[
\begin{bmatrix}
1 & 1 & 1 & : & 1 & 0 & 0 \\
2 & 3 & -1 & : & 0 & 1 & 0 \\
-4 & 7 & 4 & : & 0 & 0 & 1
\end{bmatrix}
\]

Our goal is to reduce the left-hand side to the identity matrix using elementary row operations.

### Step 1: Make the first column's elements below the pivot (in row 1, column 1) equal to zero.
- We will first use row 1 to eliminate the \( 2 \) in row 2 and the \( -4 \) in row 3.

**Row operation for row 2 (R2):**  
\( R2 \rightarrow R2 - 2 \cdot R1 \)  
\[ 
R2 = [2, 3, -1, 0, 1, 0] - 2 \times [1, 1, 1, 1, 0, 0] 
\]
\[
R2 = [0, 1, -3, -2, 1, 0]
\]

**Row operation for row 3 (R3):**  
\( R3 \rightarrow R3 + 4 \cdot R1 \)  
\[
R3 = [-4, 7, 4, 0, 0, 1] + 4 \times [1, 1, 1, 1, 0, 0] 
\]
\[
R3 = [0, 11, 8, 4, 0, 1]
\]

The updated matrix is:

\[
\begin{bmatrix}
1 & 1 & 1 & : & 1 & 0 & 0 \\
0 & 1 & -3 & : & -2 & 1 & 0 \\
0 & 11 & 8 & : & 4 & 0 & 1
\end{bmatrix}
\]

### Step 2: Eliminate the element in row 3, column 2.

**Row operation for row 3 (R3):**  
\( R3 \rightarrow R3 - 11 \cdot R2 \)
\[
R3 = [0, 11, 8, 4, 0, 1] - 11 \times [0, 1, -3, -2, 1, 0] 
\]
\[
R3 = [0, 0, 41, 26, -11, 1]
\]

The updated matrix is:

\[
\begin{bmatrix}
1 & 1 & 1 & : & 1 & 0 & 0 \\
0 & 1 & -3 & : & -2 & 1 & 0 \\
0 & 0 & 41 & : & 26 & -11 & 1
\end{bmatrix}
\]

### Step 3: Make the third row’s pivot (row 3, column 3) equal to 1.

**Row operation for row 3 (R3):**  
\( R3 \rightarrow \frac{R3}{41} \

Perform elementary row operations, A:In = [1 1 1 : 1 0 0] [2 3 -1 : 0 1 0] [-4 7 4 : 0 0 1]

Learn how to perform elementary row operations on an augmented matrix using Gaussian elimination. This detailed solution involves transforming the matrix into its identity form using row reduction. Ideal for students in Grades 10-12 studying Linear Algebra.

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