Math Problem Statement
Find the row echelon form of the given matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Solution
To solve this problem and identify the row echelon form of the given matrix , let's perform the row reduction process.
The matrix given is:
1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}$$ ### Step-by-step Row Reduction to Row Echelon Form: 1. **First Pivot Element:** The first pivot element is 1 (top-left corner). Use the first row to eliminate elements below this pivot in the first column. Perform $$R_2 \to R_2 - 4R_1$$ and $$R_3 \to R_3 - 7R_1$$: $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -12 \\ \end{bmatrix}$$ 2. **Second Pivot Element:** The pivot element in the second row is $$-3$$ (row 2, column 2). Scale row 2 by $$-\frac{1}{3}$$: $$R_2 \to \frac{R_2}{-3}$$ $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & -6 & -12 \\ \end{bmatrix}$$ 3. **Eliminate Third Row (Column 2):** Use $$R_2$$ to eliminate the second element in the third row: $$R_3 \to R_3 + 6R_2$$ $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ \end{bmatrix}$$ This is now the **row echelon form** of the matrix. ### Correct Answer: The row echelon form of the matrix is: $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ \end{bmatrix}$$ Looking at the given options, the **correct choice** is the **first option**. --- Would you like a detailed explanation of any of the steps or have additional questions? Here are 5 related questions to expand your understanding: 1. What is the difference between row echelon form and reduced row echelon form? 2. How is the row echelon form used to solve a system of linear equations? 3. What are the conditions for a matrix to be in row echelon form? 4. How does row echelon form relate to matrix rank? 5. Can all square matrices be reduced to row echelon form? **Tip:** Always check each step carefully while performing row reduction to avoid arithmetic errors, as they can drastically affect the result!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Row Reduction
Row Echelon Form
Linear Algebra
Formulas
Row Reduction Rules
Elementary Row Operations: Scaling, Row Replacement
Theorems
Row Echelon Form Definition
Suitable Grade Level
Undergraduate or Advanced High School (Grades 11-12)