Math Problem Statement
the last option is E.none of these tell me the correct option
Solution
The problem asks for expressing the matrix as a product of elementary matrices based on the given row reduction steps. The original matrix is:
0 & 2 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{pmatrix}$$ The row operations performed are: 1. $$R_1 \leftrightarrow R_2$$ (Swap row 1 and row 2) 2. $$R_3 \rightarrow R_3 - R_1$$ 3. $$R_2 \rightarrow \frac{1}{2}R_2$$ (Multiply row 2 by $$\frac{1}{2}$$) 4. $$R_3 \rightarrow R_3 - R_2$$ 5. $$R_1 \rightarrow R_1 - R_3$$ Now, let's analyze how each of these operations affects the matrix $$M$$ and its corresponding elementary matrices. ### 1. $$R_1 \leftrightarrow R_2$$ Swapping rows 1 and 2 corresponds to multiplying $$M$$ by the elementary matrix: $$E_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ ### 2. $$R_3 \rightarrow R_3 - R_1$$ Subtracting row 1 from row 3 corresponds to multiplying $$M$$ by the elementary matrix: $$E_2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix}$$ ### 3. $$R_2 \rightarrow \frac{1}{2}R_2$$ Multiplying row 2 by $$\frac{1}{2}$$ corresponds to multiplying $$M$$ by the elementary matrix: $$E_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ ### 4. $$R_3 \rightarrow R_3 - R_2$$ Subtracting row 2 from row 3 corresponds to multiplying $$M$$ by the elementary matrix: $$E_4 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$$ ### 5. $$R_1 \rightarrow R_1 - R_3$$ Subtracting row 3 from row 1 corresponds to multiplying $$M$$ by the elementary matrix: $$E_5 = \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ ### Final Expression: The product of these elementary matrices gives the overall transformation, so the matrix $$M$$ can be written as: $$A = E_5 E_4 E_3 E_2 E_1 M$$ After analyzing the options provided (A, B, C, D, and E), none of the options seem to match this exact product. Therefore, **option E (None of these)** is the correct answer. Let me know if you'd like further clarification on any of these steps or if you have additional questions! Here are five follow-up questions: 1. How do elementary matrices relate to row operations? 2. Why do row swaps correspond to non-diagonal elementary matrices? 3. What is the determinant of an elementary matrix that multiplies a row by a scalar? 4. How do row reductions affect the rank of a matrix? 5. Can you express the inverse of a matrix using elementary matrices? **Tip**: To find the inverse of a matrix, apply the same row operations used to reduce it to the identity matrix and track the changes in an identity matrix alongside.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Elementary Matrices
Row Operations
Formulas
Elementary matrix product for row operations
Matrix multiplication
Theorems
Elementary matrix theorem: Row operations as matrix products
Suitable Grade Level
College Level
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