Math Problem Statement

Given the factorization 𝐴=⎑⎣⎒⎒2βˆ’10βˆ’64βˆ’14βˆ’67⎀⎦βŽ₯βŽ₯=⎑⎣⎒⎒100-3102-41⎀⎦βŽ₯βŽ₯⎑⎣⎒⎒2-1001-1003⎀⎦βŽ₯βŽ₯,

find which elementary row operations are used to reduce 𝐴 to upper triangular form. You can answer this question without doing any computations.

  1. 𝑅2←𝑅2+ Answer 1 Question 1 ×𝑅1
  2. 𝑅3←𝑅3+ Answer 2 Question 1 ×𝑅1
  3. 𝑅3←𝑅3+ Answer 3 Question 1 ×𝑅2

Solution

To find the elementary row operations used to reduce the matrix AA to upper triangular form, we will analyze the process step-by-step based on the given factorization.

We are given the matrix AA and its factorization:

2 & -10 & -6 \\ 4 & -14 & -6 \\ 7 & 3 & 7 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 2 & -10 & 1 \end{bmatrix} \begin{bmatrix} 2 & -10 & -6 \\ 0 & -4 & -2 \\ 0 & 0 & 3 \end{bmatrix}$$ The goal is to express the row operations used in the factorization process: 1. **First row operation**: We first look at the matrix and see that the second row has a leading entry of 4. To zero out this entry (turn it into a 0), we perform the row operation: $$R_2 \leftarrow R_2 - 2R_1$$ This operation ensures that the first entry of the second row becomes 0. 2. **Second row operation**: Now, to deal with the entry in the third row under the first column (which is 7), we perform the operation: $$R_3 \leftarrow R_3 - 3R_1$$ This ensures the first entry in the third row becomes 0. 3. **Third row operation**: Finally, we need to eliminate the element in the third row, second column. Since the second entry in the second row is now -4, we apply the row operation: $$R_3 \leftarrow R_3 + \frac{5}{2} R_2$$ This operation will turn the (3,2) entry into 0, achieving upper triangular form. ### Summary of the operations: 1. $$R_2 \leftarrow R_2 - 2R_1$$ 2. $$R_3 \leftarrow R_3 - 3R_1$$ 3. $$R_3 \leftarrow R_3 + \frac{5}{2}R_2$$ Thus, the answers to the blanks are: 1. $$\text{Answer 1} = -2$$ 2. $$\text{Answer 2} = -3$$ 3. $$\text{Answer 3} = \frac{5}{2}$$ Let me know if you'd like further details or have any additional questions! Here are some related questions for further exploration: 1. How does the factorization method relate to Gaussian elimination? 2. How do elementary row operations impact the determinant of a matrix? 3. How can you use these row operations to solve a system of linear equations? 4. What happens if we used column operations instead of row operations? 5. How can we verify the correctness of these row operations through matrix multiplication? **Tip:** Always remember that for Gaussian elimination, the goal is to create zeros below the diagonal. This simplifies solving linear systems!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Factorization
Gaussian Elimination
Elementary Row Operations

Formulas

Elementary Row Operations: R2 ← R2 + cR1
Gaussian Elimination

Theorems

Gaussian Elimination Theorem
LU Factorization

Suitable Grade Level

Grades 11-12