Math Problem Statement
Solution
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 , from the options given.
Let's break it down step-by-step:
Step 1: Understanding the Transformation
We start with the matrix:
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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Elementary Matrices
Row Operations
Formulas
Elementary matrix representation of row operations
Matrix multiplication to verify echelon form transformation
Theorems
Elementary Matrices Theorem: Each elementary matrix represents a single row operation
Row Operations Theorem: Any sequence of row operations can be represented as a product of elementary matrices
Suitable Grade Level
College level or advanced high school