Math Problem Statement
Find the row-echelon form of the matrix: [[2, 3, 4], [1, -1, 2], [2, -5, 6]]
Solution
The task requires us to find the row-echelon form of the given matrix:
2 & 3 & 4 \\ 1 & -1 & 2 \\ 2 & -5 & 6 \end{bmatrix}$$ ### Step-by-Step Procedure: 1. **Make the first pivot** (top-left) 1. We can do this by dividing the first row by 2. $$\begin{bmatrix} 1 & \frac{3}{2} & 2 \\ 1 & -1 & 2 \\ 2 & -5 & 6 \end{bmatrix}$$ 2. **Eliminate the entries below the pivot**. Subtract Row 1 from Row 2 and Row 1 multiplied by 2 from Row 3. - $$R_2 = R_2 - R_1$$ - $$R_3 = R_3 - 2 \cdot R_1$$ $$\begin{bmatrix} 1 & \frac{3}{2} & 2 \\ 0 & -\frac{5}{2} & 0 \\ 0 & -\frac{11}{2} & 2 \end{bmatrix}$$ 3. **Make the second pivot** (second row, second column) a 1. We do this by dividing the second row by $$-\frac{5}{2}$$. $$\begin{bmatrix} 1 & \frac{3}{2} & 2 \\ 0 & 1 & 0 \\ 0 & -\frac{11}{2} & 2 \end{bmatrix}$$ 4. **Eliminate the entries below the second pivot**. Subtract $$\frac{11}{2} \times \text{Row 2}$$ from Row 3. $$R_3 = R_3 + \frac{11}{2} \times R_2$$ This simplifies the matrix to: $$\begin{bmatrix} 1 & \frac{3}{2} & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$$ 5. **Make the last pivot** (third row, third column) a 1. Divide Row 3 by 2. $$\begin{bmatrix} 1 & \frac{3}{2} & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ This is the **row-echelon form** of the matrix. Would you like further details on any of the steps? --- Here are 5 related questions for deeper understanding: 1. How does row-echelon form help in solving systems of linear equations? 2. What is the difference between row-echelon form and reduced row-echelon form? 3. How can you determine the rank of a matrix from its row-echelon form? 4. How does Gaussian elimination relate to row-echelon form? 5. What role do pivot elements play in transforming a matrix to row-echelon form? **Tip**: When performing row operations, always track the operations carefully, as even small errors can lead to incorrect results!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Row-Echelon Form
Gaussian Elimination
Formulas
Row operations to perform elementary row transformations
Pivot element identification and manipulation
Theorems
Gaussian Elimination
Suitable Grade Level
College-level Linear Algebra