Elementary Row Operations to Reduce Matrix A to Upper Triangular Form

Math Problem Statement

Given the factorization 𝐴=⎡⎣⎢⎢2−64−14−60−17⎤⎦⎥⎥=⎡⎣⎢⎢1−3201−4001⎤⎦⎥⎥⎡⎣⎢⎢200−1100−13⎤⎦⎥⎥,

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

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

Solution

We are given the matrix $A$ and its factorization into the product of three matrices. The factorization shows how elementary row operations are used to reduce the matrix to upper triangular form.

The given factorization is:

2 & -6 & 4 \\ -1 & 4 & -6 \\ -1 & 7 & -17 \end{bmatrix} = \begin{bmatrix} 1 & -3 & 2 \\ 0 & 1 & -4 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 3 \end{bmatrix}$$ This factorization indicates the sequence of elementary row operations that were applied to the matrix. ### Answering the row operations: 1. **$$R_2 \leftarrow R_2 + 3R_1$$**: - From the factorization, we can see that the second row in the middle matrix has become $$[0, 1, -4]$$, indicating that the second row was modified by adding a multiple of the first row. Specifically, $$R_2$$ was replaced by $$R_2 + 3R_1$$. 2. **$$R_3 \leftarrow R_3 + R_1$$**: - Similarly, the third row in the middle matrix becomes $$[0, 0, 1]$$, which shows that the third row was modified by adding $$R_1$$ to it. 3. **$$R_3 \leftarrow R_3 + R_2$$**: - After the first two operations, the third row needs to be modified again by adding a multiple of the second row. This operation is evident from the final form of the third row in the middle matrix. ### Summary of the row operations: 1. $$R_2 \leftarrow R_2 + 3R_1$$ 2. $$R_3 \leftarrow R_3 + R_1$$ 3. $$R_3 \leftarrow R_3 + R_2$$ Would you like more details on any of these steps or a breakdown of why these operations work as they do?

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Elementary Row Operations
Matrix Factorization
Upper Triangular Matrices

Formulas

Elementary Row Operations
Matrix Factorization
Upper Triangular Matrix Form

Theorems

Row Echelon Form
Elementary Row Operation Theorem

Suitable Grade Level

Grades 11-12

Related Recommendation

Elementary Row Operations to Reduce Matrix to Upper Triangular Form

Matrix Row Operations and Elementary Matrices

Matrix Row Operations to Upper Triangular Form

Triangular Form Reduction using Gaussian Elimination for 3x3 Matrix

Convert a 4x4 Matrix to Upper Triangular Form Using Gaussian Elimination