Math Problem Statement

Solve using RREF: Solve the system of equations using matrix. x1 + x3 = 5, x1 + x2 + 2x3 =-2, x1 +2x2 +3x3 =1

Solution

To solve the given system of equations using the Reduced Row Echelon Form (RREF), we represent the system as an augmented matrix and perform row operations. The system is:

1. \quad x_1 + x_3 &= 5 \\ 2. \quad x_1 + x_2 + 2x_3 &= -2 \\ 3. \quad x_1 + 2x_2 + 3x_3 &= 1$$ ### Step 1: Write the augmented matrix $$\begin{bmatrix} 1 & 0 & 1 & 5 \\ 1 & 1 & 2 & -2 \\ 1 & 2 & 3 & 1 \end{bmatrix}$$ ### Step 2: Perform row operations to reduce the matrix to RREF I'll perform the computations to find the RREF. sympy as sp # Define the augmented matrix matrix = sp.Matrix([ [1, 0, 1, 5], [1, 1, 2, -2], [1, 2, 3, 1] ]) # Perform row reduction to RREF rref_matrix, pivots = matrix.rref() rref_matrixThe Reduced Row Echelon Form (RREF) of the augmented matrix is: $$\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ ### Interpretation of the RREF From the matrix, the system of equations becomes: 1. $$x_1 + x_3 = 0$$ 2. $$x_2 + x_3 = 0$$ 3. $$0 = 1$$ (this is inconsistent) Since the third row represents a contradiction ($$0 = 1$$), the system of equations is inconsistent, meaning there are no solutions. --- Would you like a detailed breakdown of each step? Here are some follow-up questions to consider: 1. What does it mean for a system of equations to be inconsistent? 2. How are row operations applied systematically in matrix reduction? 3. Can a system have infinitely many solutions? When does that occur? 4. How can we verify matrix calculations manually? 5. What are the differences between RREF and REF? **Tip:** Always double-check for arithmetic errors when performing row operations manually!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Operations
Row Echelon Form (REF)
Reduced Row Echelon Form (RREF)

Formulas

Row Echelon Form (REF)
Reduced Row Echelon Form (RREF)

Theorems

Consistency of Systems of Linear Equations
Gaussian Elimination
Row Operations

Suitable Grade Level

Grades 10-12