Math Problem Statement
Reduce the matrix \( A = \begin{bmatrix} -1 & -2 & -2 & -4 \\ 1 & 3 & 3 & 7 \\ 1 & -3 & -2 & -7 \end{bmatrix} \) to reduced row-echelon form.
Solution
The task is to reduce the given matrix to its reduced row-echelon form (RREF). Let me solve it step-by-step.
The matrix is:
-1 & -2 & -2 & -4 \\ 1 & 3 & 3 & 7 \\ 1 & -3 & -2 & -7 \end{bmatrix}.$$ ### Step 1: Perform row operations to achieve the RREF 1. Make the first pivot (top-left element) equal to 1. Divide the first row $$R_1$$ by $$-1$$: $$R_1 \to \frac{R_1}{-1}.$$ The matrix becomes: $$\begin{bmatrix} 1 & 2 & 2 & 4 \\ 1 & 3 & 3 & 7 \\ 1 & -3 & -2 & -7 \end{bmatrix}.$$ 2. Eliminate the first column entries below the pivot (first column in $$R_2$$ and $$R_3$$). Perform: $$R_2 \to R_2 - R_1, \quad R_3 \to R_3 - R_1.$$ After performing the row operations: $$\begin{bmatrix} 1 & 2 & 2 & 4 \\ 0 & 1 & 1 & 3 \\ 0 & -5 & -4 & -11 \end{bmatrix}.$$ 3. Make the second pivot (second column of $$R_2$$) equal to 1. This pivot is already 1, so no change is needed. 4. Eliminate the second column entries in $$R_1$$ and $$R_3$$. Perform: $$R_1 \to R_1 - 2R_2, \quad R_3 \to R_3 + 5R_2.$$ After these row operations: $$\begin{bmatrix} 1 & 0 & 0 & -2 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 1 & 4 \end{bmatrix}.$$ 5. Eliminate the third column entry in $$R_2$$. Perform: \[ R_2 \to R_2Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Row Reduction
Reduced Row-Echelon Form
Formulas
Row reduction operations (row addition, scalar multiplication, row swapping)
Theorems
Properties of row-echelon form and reduced row-echelon form
Suitable Grade Level
Grades 11-12 or Early College