Math Problem Statement
Find the reduced row echelon form of this augmented matrix
Solution
Let's solve the given problem. The augmented matrix provided is:
-2 & 3 & 0 & 350 \\ -13 & 19 & 0 & 100 \\ 6 & -9 & 1 & 150 \end{bmatrix}$$ We aim to transform it into its **reduced row echelon form (RREF)**. ### Steps for RREF: 1. Use row operations to create leading ones in each row. 2. Ensure each leading one is the only non-zero entry in its column. 3. Work row by row to eliminate entries below and above each pivot. I will calculate the reduced row echelon form.import sympy as sp # Define the augmented matrix matrix = sp.Matrix([ [-2, 3, 0, 350], [-13, 19, 0, 100], [6, -9, 1, 150] ]) # Compute the Reduced Row Echelon Form (RREF) rref_matrix, pivot_columns = matrix.rref() rref_matrixMatrix([ [1, 0, 0, 6350], [0, 1, 0, 4350], [0, 0, 1, 1200]])Matrix([ [1, 0, 0, 6350], [0, 1, 0, 4350], [0, 0, 1, 1200]])successThe reduced row echelon form (RREF) of the given augmented matrix is: $$\begin{bmatrix} 1 & 0 & 0 & 6350 \\ 0 & 1 & 0 & 4350 \\ 0 & 0 & 1 & 1200 \end{bmatrix}$$ ### Interpretation: - The matrix is now fully simplified, indicating the solutions for the system of equations represented by the original augmented matrix. Let me know if you'd like further clarification or additional steps explained! ### Questions to Explore: 1. What are the implications of having each row reduced to a unique pivot variable? 2. How do row operations affect the determinant of a matrix? 3. How can the reduced matrix form help in determining system consistency? 4. What if the augmented matrix had infinitely many solutions? How would RREF look? 5. Can we generalize the process of row reduction for higher dimensions? ### Tip: When performing RREF, always start with the leftmost column and ensure pivot elements are non-zero by swapping rows if necessary.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Row Reduction
Augmented Matrix
Reduced Row Echelon Form (RREF)
Formulas
Row Operations for RREF
Theorems
Reduced Row Echelon Form (RREF) Theorem
Suitable Grade Level
College-level or advanced high school