https://file.aiquickdraw.com/mathbot/file/1728362937533juwpx35y.jpg

The problem provided involves identifying elementary matrices used in a series of row operations to convert a matrix into its echelon form. The given echelon form transformation is presented in several steps, each corresponding to a row operation. The goal is to determine the elementary matrix corresponding to the first row operation, denoted as \(E_1\), from the options given.

Let's break it down step-by-step:

### Step 1: Understanding the Transformation
We start with the matrix:
\[
\begin{bmatrix}
1 & 3 & 1 & 3 \\
2 & 1 & -1 & -1 \\
1 & 2 & 0 & -2
\end{bmatrix}
\]
The first transformation, denoted as \( (1) \), is performed by multiplying the second row by -2 and adding it to the third row (a row replacement operation):
\[
R_3 \to R_3 - 2R_2
\]
This gives the new matrix:
\[
\begin{bmatrix}
1 & 3 & 1 & 3 \\
2 & 1 & -1 & -1 \\
0 & -1 & 2 & 4
\end{bmatrix}
\]

### Step 2: Elementary Matrix for \(E_1\)
An elementary matrix represents the row operation applied to the identity matrix. The operation \( R_3 \to R_3 - 2R_2 \) corresponds to modifying the identity matrix by performing the same operation:
\[
E_1 = 
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-2 & 0 & 1
\end{bmatrix}
\]

Looking at the two options given for \(E_1\), the correct elementary matrix is:
\[
\boxed{
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-2 & 0 & 1
\end{bmatrix}
}
\]

Would you like further details on any steps or explanations? Here are some questions to deepen your understanding:

1. How is an elementary matrix constructed from row operations?
2. What types of row operations correspond to different elementary matrices?
3. What does multiplying a matrix by an elementary matrix accomplish?
4. Can row swapping be represented by an elementary matrix, and how?
5. What role do elementary matrices play in finding the inverse of a matrix?

**Tip**: Each row operation (scaling, swapping, or adding multiples of rows) directly corresponds to a unique elementary matrix.

Identify each elementary matrix for the echelon form transformation shown in the image. Specifically, identify E1.

Learn how to identify the elementary matrices used in transforming a matrix to its echelon form. The step-by-step breakdown shows how to apply row operations and recognize their corresponding elementary matrices, particularly for E1.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation