Math Problem Statement

Consider the following matrix:

𝐴=⎡⎣⎢⎢2−64−14−60−17⎤⎦⎥⎥.

We will apply row operations to 𝐴 to make it upper triangular. After our first row operation, we find

𝐵=⎡⎣⎢⎢204−11−60−17⎤⎦⎥⎥.

To arrive at 𝐵, we replaced 𝑅2 of 𝐴 with 𝑅2+ Answer 1 Question 2 ×𝑅1.

We then apply another row operation to 𝐵 to find

𝐶=⎡⎣⎢⎢200−11−40−17⎤⎦⎥⎥.

To arrive at 𝐶, we replaced 𝑅3 of 𝐵 with 𝑅3+ Answer 2 Question 2 ×𝑅1.

We apply a final row operation to 𝐶 to find

𝐷=⎡⎣⎢⎢200−1100−13⎤⎦⎥⎥.

To arrive at 𝐷, we replaced 𝑅3 of 𝐶 with 𝑅3+ Answer 3 Question 2 ×𝑅2.

We now let 𝐸1 be the elementary matrix representing the first row operation, i.e. 𝐵=𝐸1𝐴. Then,

(2,1)-entry of 𝐸1: Answer 4 Question 2

(3,1)-entry of 𝐸1: Answer 5 Question 2

(3,2)-entry of 𝐸1: Answer 6 Question 2

Let 𝐸2 be the elementary matrix representing the second row operation, i.e. 𝐶=𝐸2𝐵. Then,

(2,1)-entry of 𝐸2: Answer 7 Question 2

(3,1)-entry of 𝐸2: Answer 8 Question 2

(3,2)-entry of 𝐸2: Answer 9 Question 2

Let 𝐸3 be the elementary matrix representing the third row operation, i.e. 𝐷=𝐸3𝐶. Then,

(2,1)-entry of 𝐸3: Answer 10 Question 2

(3,1)-entry of 𝐸3: Answer 11 Question 2(3,2)-entry of 𝐸3: Answer 12 Question 2

Solution

We are given a matrix AA and a sequence of row operations that lead to new matrices. Let’s break down each step and find the required values.

Initial Matrix AA:

A=[2−64−14−60−17]A = \begin{bmatrix} 2 & -6 & 4 \\ -1 & 4 & -6 \\ 0 & -1 & 7 \end{bmatrix}

Row Operation 1:

To obtain matrix BB, we perform the row operation on row 2: R2→R2+2R1R_2 \rightarrow R_2 + 2R_1. This means we add 2 times the first row to the second row. Let’s compute:

R2=(−1,4,−6)+2×(2,−6,4)=(−1+4,4−12,−6+8)=(3,−8,2)R_2 = (-1, 4, -6) + 2 \times (2, -6, 4) = (-1 + 4, 4 - 12, -6 + 8) = (3, -8, 2)

Thus, matrix BB is:

B=[2−643−820−17]B = \begin{bmatrix} 2 & -6 & 4 \\ 3 & -8 & 2 \\ 0 & -1 & 7 \end{bmatrix}

Answer 1 (First Row Operation):

We replaced R2R_2 of AA with R2+2R1R_2 + 2R_1, so the multiplier for the first row operation is 2.

Row Operation 2:

Next, to obtain matrix CC, we perform

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Math Problem Analysis

Mathematical Concepts

Matrix Operations
Row Operations
Elementary Matrices
Linear Algebra

Formulas

Row operations for matrix transformations
Elementary matrices representing row operations

Theorems

Gaussian elimination
Matrix multiplication

Suitable Grade Level

Grades 11-12

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