Math Problem Statement
Row echelon form of matrix [0 -1 0 1 1 -1 3 2 3]
Solution
To find the row echelon form (REF) of the matrix:
0 & -1 & 0 \\ 1 & 1 & -1 \\ 3 & 2 & 3 \end{bmatrix}$$ We will apply the following steps in Gaussian elimination: ### Step 1: Swap rows to get a non-zero element in the top-left corner The element at position (1,1) is 0, so let's swap Row 1 and Row 2. This gives us: $$\begin{bmatrix} 1 & 1 & -1 \\ 0 & -1 & 0 \\ 3 & 2 & 3 \end{bmatrix}$$ ### Step 2: Eliminate the element below the pivot (position 2,1) We now use Row 1 (pivot 1) to eliminate the element in position (3,1). To do this, replace Row 3 by subtracting 3 times Row 1 from Row 3: $$R_3 \to R_3 - 3R_1$$ So, Row 3 becomes: \[ [3 - 3(1), 2 - 3(1), 3 - 3(-1)] = [0, -1, 6] \] This gives us the updated matrix: $$\begin{bmatrix} 1 & 1 & -1 \\ 0 & -1 & 0 \\ 0 & -1 & 6 \end{bmatrix}$$ ### Step 3: Eliminate the element below the pivot in column 2 Next, we use Row 2 (pivot -1) to eliminate the element in position (3,2). We replace Row 3 by subtracting Row 2 from Row 3: $$R_3 \to R_3 - R_2$$ Row 3 becomes: \[ [0 - 0, -1 - (-1), 6 - 0] = [0, 0, 6] \] Now, the matrix looks like this: $$\begin{bmatrix} 1 & 1 & -1 \\ 0 & -1 & 0 \\ 0 & 0 & 6 \end{bmatrix}$$ ### Step 4: Make the pivots equal to 1 (optional, for reduced row echelon form) To make the pivot in the last row equal to 1, divide Row 3 by 6: $$R_3 \to \frac{1}{6} R_3$$ Row 3 becomes: $$[0, 0, 1]$$ Thus, the final row echelon form of the matrix is: $$\begin{bmatrix} 1 & 1 & -1 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ ### Summary: The **row echelon form** (REF) of the given matrix is: $$\begin{bmatrix} 1 & 1 & -1 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ Would you like further clarification on any of these steps, or do you have any other questions? Here are some related questions you might find interesting: 1. How do you convert a matrix to reduced row echelon form (RREF)? 2. What is the significance of row echelon form in solving linear systems? 3. How do you perform Gaussian elimination in a larger matrix? 4. Can a matrix have more than one row echelon form? 5. What is the difference between row echelon form and column echelon form? **Tip:** Always start by moving non-zero elements to the pivot positions in the first column to simplify the elimination process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Row Echelon Form
Gaussian Elimination
Formulas
Row Echelon Form
Gaussian elimination method
Theorems
Gaussian elimination
Suitable Grade Level
Grades 10-12