Math Problem Statement
0 −4 2 2 1 0 0 5 −3 −1 1 −4 2 4 1 0 8 2 2 7 0 4 −3 0 2 do ref
Solution
To perform row echelon form (REF) on the matrix:
0 & -4 & 2 & 2 & 1 \\ 0 & 5 & -3 & -1 & 1 \\ -4 & 2 & 4 & 1 & 0 \\ 8 & 2 & 2 & 7 & 0 \\ 4 & -3 & 0 & 2 & -3 \\ \end{pmatrix}$$ we will perform a series of row operations to transform it into row echelon form. Here are the steps: ### Step 1: Swap Rows Since the first element in the first row is zero, we need to swap rows to place a non-zero element at the top left. Let's swap Row 1 with Row 3: $$\begin{pmatrix} -4 & 2 & 4 & 1 & 0 \\ 0 & 5 & -3 & -1 & 1 \\ 0 & -4 & 2 & 2 & 1 \\ 8 & 2 & 2 & 7 & 0 \\ 4 & -3 & 0 & 2 & -3 \\ \end{pmatrix}$$ ### Step 2: Make the Leading Entry in Row 1 a 1 To make the leading entry in Row 1 a 1, divide Row 1 by -4: $$\begin{pmatrix} 1 & -0.5 & -1 & -0.25 & 0 \\ 0 & 5 & -3 & -1 & 1 \\ 0 & -4 & 2 & 2 & 1 \\ 8 & 2 & 2 & 7 & 0 \\ 4 & -3 & 0 & 2 & -3 \\ \end{pmatrix}$$ ### Step 3: Eliminate the Entries Below the Leading 1 in Column 1 - Row 4: Replace Row 4 with Row 4 - 8 * Row 1 - Row 5: Replace Row 5 with Row 5 - 4 * Row 1 After these operations, we get: $$\begin{pmatrix} 1 & -0.5 & -1 & -0.25 & 0 \\ 0 & 5 & -3 & -1 & 1 \\ 0 & -4 & 2 & 2 & 1 \\ 0 & 6 & 10 & 9 & 0 \\ 0 & -1 & 4 & 3 & -3 \\ \end{pmatrix}$$ ### Step 4: Make the Leading Entry in Row 2 a 1 Divide Row 2 by 5 to make the leading entry in Row 2 a 1: \[ \begin{pmatrix} 1 & -0.5 & -1 & -0.25 & 0 \\ 0 & 1 & -0.6 & -0.2 & 0.2 \\ 0 & -4 & 2 & 2 & 1 \\ 0 & 6 & 10 &Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Row Echelon Form
Gaussian Elimination
Formulas
Row operations: swap, multiply, and add rows
Theorems
Row Echelon Form Transformation
Suitable Grade Level
Grades 10-12
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