Math Problem Statement
Find an elementary matrix E so that B = EA, where A = [[8, 4, 4], [0, 2, 8], [4, 0, 0]] and B = [[8, 4, 4], [4, 0, 0], [0, 2, 8]].
Solution
The problem asks to find an elementary matrix such that , where the matrices and are given as:
8 & 4 & 4 \\ 0 & 2 & 8 \\ 4 & 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 8 & 4 & 4 \\ 4 & 0 & 0 \\ 0 & 2 & 8 \end{pmatrix}$$ ### Approach: Elementary matrices represent row operations, so $$E$$ is the matrix that transforms matrix $$A$$ into matrix $$B$$ through row operations. By comparing $$A$$ and $$B$$, we observe that the second and third rows of $$A$$ have been swapped to form $$B$$. Therefore, the elementary matrix $$E$$ will be the one corresponding to a row swap operation between the second and third rows. ### Elementary matrix for row swap: To swap the second and third rows, the elementary matrix $$E$$ is: $$E = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$$ ### Verification: To verify, we multiply $$E$$ by $$A$$: $$EA = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 8 & 4 & 4 \\ 0 & 2 & 8 \\ 4 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 8 & 4 & 4 \\ 4 & 0 & 0 \\ 0 & 2 & 8 \end{pmatrix} = B$$ Thus, $$E$$ correctly transforms $$A$$ into $$B$$. Let me know if you'd like more details or have any questions! ### Related Questions: 1. What are the different types of elementary row operations? 2. How does matrix multiplication work with elementary matrices? 3. Can an elementary matrix have fractional or negative entries? 4. How do we know when a matrix is invertible using elementary matrices? 5. How can we use elementary matrices to find the inverse of a matrix? ### Tip: Elementary matrices are always invertible, and their inverse corresponds to undoing the row operation they represent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Elementary Matrices
Matrix Multiplication
Formulas
B = EA
Row swap elementary matrix
Theorems
Elementary matrix transforms one matrix into another through row operations.
Suitable Grade Level
Undergraduate (Linear Algebra)