Math Problem Statement
find e1 using guassian-jordan method
Solution
In this image, the matrix is transformed into through the application of an elementary row operation as part of the Gaussian-Jordan method.
We need to find the elementary matrix , which performs this row operation. Let's analyze the transition from matrix to .
The matrix is:
1 & -6 & 0 \\ -21 & 0 & 3 \\ 0 & 1 & 0 \end{bmatrix}$$ The resulting matrix $$E_1 A$$ is: $$E_1 A = \begin{bmatrix} 1 & 0 & 0 \\ -21 & 0 & 3 \\ 0 & 1 & 0 \end{bmatrix}$$ ### Step-by-step process using Gaussian-Jordan method: 1. **Look at the first row transformation**: We see that row 1 in matrix $$A$$ has changed from $$[1, -6, 0]$$ to $$[1, 0, 0]$$ in matrix $$E_1 A$$. This means that a row operation was applied to set the second element of the first row to 0. 2. **Determine the elementary row operation**: To transform the first row of matrix $$A$$, the row operation must have been to add 6 times the third row to the first row (i.e., $$R_1 \rightarrow R_1 + 6R_3$$). Thus, $$E_1$$, the elementary matrix corresponding to this row operation, would be the identity matrix with the appropriate transformation applied to the first row: $$E_1 = \begin{bmatrix} 1 & 6 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ ### Verification: Multiplying $$E_1$$ by $$A$$: $$E_1 A = \begin{bmatrix} 1 & 6 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & -6 & 0 \\ -21 & 0 & 3 \\ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ -21 & 0 & 3 \\ 0 & 1 & 0 \end{bmatrix}$$ This confirms that $$E_1$$ is correct. Would you like more details or have any questions? Here are some related questions to deepen your understanding: 1. What are the key differences between Gaussian elimination and Gaussian-Jordan method? 2. How does the form of an elementary matrix change based on the row operation applied? 3. Why is Gaussian-Jordan elimination used for finding the inverse of a matrix? 4. What happens if we apply row swaps instead of row additions in elementary matrices? 5. Can $$E_1$$ be applied to any matrix or is it specific to matrix $$A$$? **Tip:** When performing Gaussian elimination, always track row operations carefully, as they directly determine the corresponding elementary matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Elementary Matrices
Gaussian-Jordan Elimination
Formulas
R1 -> R1 + 6R3
E_1 A = Identity matrix after row operations
Theorems
Gaussian Elimination
Elementary Matrix Theorem
Suitable Grade Level
Undergraduate Level
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