Math Problem Statement
Apply elimination to the 2 by 3 augmented matrix [A b]. What is the triangular system Ux = c? What is the solution x?
Solution
Let's analyze the system of equations and solve it step by step.
The augmented matrix is given by:
-1 & -2 & 2 \\ 1 & 0 & 2 \end{pmatrix}$$ We want to apply Gaussian elimination to convert this into an upper triangular matrix $$U$$, and then back-substitute to find the solution $$x$$. ### Step 1: Apply elimination We can start by using row operations to create zeros below the diagonal. **Row 1**: $$[-1 \quad -2 \quad 2]$$ **Row 2**: $$[1 \quad 0 \quad 2]$$ To eliminate the first element of Row 2, we add Row 1 to Row 2: \[ R_2 = R_2 + R_1 = [1 + (-1) \quad 0 + (-2) \quad 2 + 2] = [0 \quad -2 \quad 4] \] So, the upper triangular matrix $$U$$ becomes: $$U = \begin{pmatrix} -1 & -2 \\ 0 & -2 \end{pmatrix}$$ The corresponding constant vector $$c$$ is: $$c = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$$ ### Step 2: Back-substitution Now we solve the system $$Ux = c$$. From the second row: $$-2x_2 = 4 \quad \Rightarrow \quad x_2 = \frac{4}{-2} = -2$$ Substitute $$x_2 = -2$$ into the first equation: $$-1x_1 - 2(-2) = 2 \quad \Rightarrow \quad -x_1 + 4 = 2$$ $$-x_1 = 2 - 4 = -2 \quad \Rightarrow \quad x_1 = 2$$ Thus, the solution is: $$x = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$$ ### Summary of results: - $$U = \begin{pmatrix} -1 & -2 \\ 0 & -2 \end{pmatrix}$$ - $$c = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$$ - $$x = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$$ Would you like more details or further clarification on any part? --- Here are 5 related questions: 1. How does Gaussian elimination work in larger matrices? 2. What is the role of back-substitution in solving linear systems? 3. Can Gaussian elimination be applied to non-square matrices? 4. How do row operations affect the determinant of a matrix? 5. What happens if a matrix becomes singular during elimination? **Tip**: When performing Gaussian elimination, always ensure to pivot correctly if you encounter zeros in the diagonal to avoid division errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Matrix Operations
Formulas
Row Reduction
Back-substitution
Ax = b
Upper Triangular Matrix Ux = c
Theorems
Gaussian Elimination Theorem
System of Linear Equations
Suitable Grade Level
Undergraduate or High School (Grades 11-12)
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