Math Problem Statement
Transform this matrix to reduced row echelon form:
2 −1 −4 7 0 −30 20 −50 0 0 44 −44
1 0 0 0 1 0 0 0 1
Solution
To transform the given matrix to reduced row echelon form (RREF), we proceed step-by-step using Gaussian elimination and back-substitution. The matrix is:
2 & -1 & -4 & 7 \\ 0 & -30 & 20 & -50 \\ 0 & 0 & 44 & -44 \end{bmatrix}$$ We aim to achieve: $$\begin{bmatrix} 1 & 0 & 0 & \ast \\ 0 & 1 & 0 & \ast \\ 0 & 0 & 1 & \ast \end{bmatrix}$$ ### Step 1: Scale the first row Divide the first row by 2 to make the pivot in the first column equal to 1: $$\begin{bmatrix} 1 & -0.5 & -2 & 3.5 \\ 0 & -30 & 20 & -50 \\ 0 & 0 & 44 & -44 \end{bmatrix}$$ ### Step 2: Eliminate below the first pivot The first pivot is already 1, and the rows below already have 0 in the first column, so no further elimination is required for the first column. ### Step 3: Scale the second row Divide the second row by $$-30$$ to make the second pivot 1: $$\begin{bmatrix} 1 & -0.5 & -2 & 3.5 \\ 0 & 1 & -2/3 & 5/3 \\ 0 & 0 & 44 & -44 \end{bmatrix}$$ ### Step 4: Eliminate above and below the second pivot The second pivot is 1. Adjust row 1 to eliminate the second column entry: Row 1: $$R_1 = R_1 + 0.5 \cdot R_2$$ $$\begin{bmatrix} 1 & 0 & -8/3 & 7/3 \\ 0 & 1 & -2/3 & 5/3 \\ 0 & 0 & 44 & -44 \end{bmatrix}$$ No need to adjust row 3 for the second column, as it is already 0. ### Step 5: Scale the third row Divide the third row by 44 to make the third pivot 1: $$\begin{bmatrix} 1 & 0 & -8/3 & 7/3 \\ 0 & 1 & -2/3 & 5/3 \\ 0 & 0 & 1 & -1 \end{bmatrix}$$ ### Step 6: Eliminate above the third pivot Adjust rows 1 and 2 to eliminate the third column entries: Row 1: $$R_1 = R_1 + \frac{8}{3} \cdot R_3$$ Row 2: $$R_2 = R_2 + \frac{2}{3} \cdot R_3$$ $$\begin{bmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \end{bmatrix}$$ This is the **reduced row echelon form (RREF)** of the matrix. ### Final RREF: $$\begin{bmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \end{bmatrix}$$ Let me know if you need step-by-step verification of these computations or any additional clarifications! --- **5 Related Questions:** 1. How can Gaussian elimination be applied to larger matrices? 2. What is the difference between row echelon form and reduced row echelon form? 3. Can every matrix be reduced to RREF? Why or why not? 4. How do you interpret the solution set of a system represented by a matrix in RREF? 5. How would you use RREF to solve a system of linear equations? **Tip:** Always check for computational accuracy in each step of row transformations to avoid propagating errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Gaussian Elimination
Reduced Row Echelon Form (RREF)
Formulas
Row transformation rules: scaling (R_i -> k * R_i), replacement (R_i -> R_i + k * R_j), and swapping rows
Division of a row by its pivot element to make the pivot 1
Theorems
Gaussian Elimination
Matrix Row Reduction
Suitable Grade Level
Grades 10-12 or College Level