Math Problem Statement
FindtherowechelonandreducedrowechelonformofthematrixA= 2 −2 4 −2 2 1 10 7 −4 4 −8 4 4 −1 14 6
Solution
Let's work through the row echelon form (REF) and reduced row echelon form (RREF) of the given matrix:
2 & -2 & 4 & -2 \\ 2 & 1 & 10 & 7 \\ -4 & 4 & -8 & 4 \\ 4 & -1 & 14 & 6 \end{bmatrix}$$ ### Step 1: Row Echelon Form (REF) The goal is to transform the matrix into an upper triangular form where all the leading coefficients (the first non-zero entry in each row) are 1, and each leading entry is to the right of the leading entry in the row above it. #### Step 1.1: Row Operations Start by making the first element of the first row a 1 by dividing the first row by 2: $$R_1 \rightarrow \frac{1}{2} R_1$$ $$A = \begin{bmatrix} 1 & -1 & 2 & -1 \\ 2 & 1 & 10 & 7 \\ -4 & 4 & -8 & 4 \\ 4 & -1 & 14 & 6 \end{bmatrix}$$ #### Step 1.2: Eliminate entries below the pivot (first element of the first row) We now use the first row to eliminate the first element of the second, third, and fourth rows: $$R_2 \rightarrow R_2 - 2R_1, \quad R_3 \rightarrow R_3 + 4R_1, \quad R_4 \rightarrow R_4 - 4R_1$$ After applying these operations: $$A = \begin{bmatrix} 1 & -1 & 2 & -1 \\ 0 & 3 & 6 & 9 \\ 0 & 0 & 0 & 0 \\ 0 & 3 & 6 & 10 \end{bmatrix}$$ #### Step 1.3: Make the pivot in the second row a 1 Divide the second row by 3: $$R_2 \rightarrow \frac{1}{3} R_2$$ $$A = \begin{bmatrix} 1 & -1 & 2 & -1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 3 & 6 & 10 \end{bmatrix}$$ #### Step 1.4: Eliminate the 3 in the fourth row Use the second row to eliminate the 3 in the second position of the fourth row: $$R_4 \rightarrow R_4 - 3R_2$$ $$A = \begin{bmatrix} 1 & -1 & 2 & -1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ ### Step 2: Reduced Row Echelon Form (RREF) To convert the matrix into reduced row echelon form, we now focus on making the leading coefficients 1 in each row and ensuring all other elements in the column of a leading 1 are zero. #### Step 2.1: Eliminate the -1 in the first row Use the second row to eliminate the -1 in the first row: $$R_1 \rightarrow R_1 + R_2$$ $$A = \begin{bmatrix} 1 & 0 & 2 & 2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ #### Step 2.2: Eliminate the 2 in the first row Use the fourth row to eliminate the 2 in the first row: $$R_1 \rightarrow R_1 - 2R_4$$ $$A = \begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ ### Final Answer - **Row Echelon Form (REF)**: $$\begin{bmatrix} 1 & -1 & 2 & -1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ - **Reduced Row Echelon Form (RREF)**: $$\begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ --- Would you like to see more detailed steps or explanations on any part? Here are some questions to explore further: 1. How do you use row operations to solve a system of linear equations? 2. Can you explain the concept of rank in relation to row echelon form? 3. What are the differences between row echelon form and reduced row echelon form? 4. How do you apply Gaussian elimination to a matrix? 5. How does the concept of pivots relate to the row echelon form? **Tip**: Always check for zero rows when simplifying to row echelon form, as they may not contribute to the rank or leading entries.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Row Echelon Form
Reduced Row Echelon Form
Gaussian Elimination
Formulas
Row Operations
Gaussian Elimination
Theorems
Row Echelon Form Theorem
Reduced Row Echelon Form Theorem
Suitable Grade Level
Grades 10-12