Math Problem Statement
Solution
From the image, the problem involves selecting the correct row operation used to transform the matrix on the left into the matrix on the right. Let's analyze the situation step by step.
Initial Matrix:
1 & 1 & -1 & | & 5 \\ 5 & 0 & -20 & | & 15 \\ -2 & 2 & 3 & | & 10 \end{bmatrix}$$ ### Transformed Matrix: $$\begin{bmatrix} 1 & 1 & -1 & | & 5 \\ 5 & 0 & -20 & | & 15 \\ 0 & 4 & 1 & | & 20 \end{bmatrix}$$ ### Observing Changes: The first two rows ($$R_1$$ and $$R_2$$) remain unchanged. The operation modifies $$R_3$$, which initially was: $$R_3 = [-2, 2, 3, | 10]$$ and becomes: $$R_3 = [0, 4, 1, | 20]$$ ### Test Each Option: 1. **Option (a): $$-R_1 + 3R_3 \to R_3$$** Performing this: \[ -R_1 = [-1, -1, 1, | -5], \quad 3R_3 = [-6, 6, 9, | 30] \] Adding: \[ -R_1 + 3R_3 = [-7, 5, 10, | 25] \quad \neq [0, 4, 1, | 20] \] Incorrect. 2. **Option (b): $$2R_1 + R_3 \to R_3$$** Performing this: \[ 2R_1 = [2, 2, -2, | 10], \quad R_3 = [-2, 2, 3, | 10] \] Adding: $$2R_1 + R_3 = [0, 4, 1, | 20]$$ Correct! 3. **Option (c): $$2R_3 - 3R_1 \to R_3$$** Performing this: \[ 2R_3 = [-4, 4, 6, | 20], \quad -3R_1 = [-3, -3, 3, | -15] \] Adding: \[ 2R_3 - 3R_1 = [-7, 1, 9, | 5] \quad \neq [0, 4, 1, | 20] \] Incorrect. 4. **Option (d): $$-3R_2 + R_3 \to R_3$$** Performing this: \[ -3R_2 = [-15, 0, 60, | -45], \quad R_3 = [-2, 2, 3, | 10] \] Adding: \[ -3R_2 + R_3 = [-17, 2, 63, | -35] \quad \neq [0, 4, 1, | 20] \] Incorrect. ### Final Answer: **(b): $$2R_1 + R_3 \to R_3$$** Would you like a detailed explanation of any step? Here are related questions for deeper understanding: 1. How do row operations affect the determinant of a matrix? 2. What are common row reduction strategies for solving systems of equations? 3. How can augmented matrices be used to find the inverse of a matrix? 4. What are the properties of elementary row operations? 5. How does Gaussian elimination relate to row operations? **Tip:** Always verify row operations by substituting values to ensure the transformation is correct.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Row Operations
Formulas
Elementary row operation: cR_i + R_j → R_j
Gaussian elimination rules for matrix transformation
Theorems
Properties of Elementary Row Operations
Suitable Grade Level
Undergraduate (Linear Algebra)
Related Recommendation
Elementary Row Operations on Augmented Matrix [1 1 1 : 1 0 0] [2 3 -1 : 0 1 0] [-4 7 4 : 0 0 1]
Understanding Allowable Actions in Gaussian Elimination
Step-by-Step Solution to Matrix Operations Using Gaussian Elimination
Matrix Row Echelon and Reduction Example
Elementary Row Operations on Matrix for Zeroing First Column Elements