Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Matrix Row Operations: Replace \(R_2\) by \((-8)R_1 + R_2\)
In the process of performing row operations to simplify a matrix, we have the following steps:
**Given Matrix:**
\[
\begin{bmatrix}
1 & 4 & -8 & -7 \\
8 & 4 & 7 & 8 \\
-5 & 1 & 5 & 2
\end{bmatrix}
\]
We aim to replace the second row (\(R_2\)) with the result of \((-8)R_1 + R_2\), where \(R_1\) is the first row and \(R_2\) is the second row. This operation is written as:
\[
R_2 \leftarrow (-8)R_1 + R_2
\]
### Steps to follow:
1. **Scaling the First Row (\(R_1\)) by -8**:
\[
(-8)R_1 = -8 \cdot \begin{bmatrix}
1 & 4 & -8 & -7
\end{bmatrix} = \begin{bmatrix}
-8 & -32 & 64 & 56
\end{bmatrix}
\]
2. **Adding the Scaled First Row to the Second Row (\(R_2\))**:
\[
R_2 = R_2 + (-8)R_1
\]
\[
\begin{bmatrix}
8 & 4 & 7 & 8
\end{bmatrix} + \begin{bmatrix}
-8 & -32 & 64 & 56
\end{bmatrix} = \begin{bmatrix}
0 & -28 & 71 & 64
\end{bmatrix}
\]
- For the 1st element: \(8 + (-8) = 0\)
- For the 2nd element: \(4 + (-32) = -28\)
- For the 3rd element: \(7 + 64 = 71\)
- For the 4th element: \(8 + 56 = 64\)
3. **Form the New Matrix with the Updated Second Row**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe306f2a1-f925-4c9a-bf5c-53ad4660f15e%2Feb0ac556-1357-43f0-bd38-8d3dcb16213c%2F61y4uvu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Row Operations: Replace \(R_2\) by \((-8)R_1 + R_2\)
In the process of performing row operations to simplify a matrix, we have the following steps:
**Given Matrix:**
\[
\begin{bmatrix}
1 & 4 & -8 & -7 \\
8 & 4 & 7 & 8 \\
-5 & 1 & 5 & 2
\end{bmatrix}
\]
We aim to replace the second row (\(R_2\)) with the result of \((-8)R_1 + R_2\), where \(R_1\) is the first row and \(R_2\) is the second row. This operation is written as:
\[
R_2 \leftarrow (-8)R_1 + R_2
\]
### Steps to follow:
1. **Scaling the First Row (\(R_1\)) by -8**:
\[
(-8)R_1 = -8 \cdot \begin{bmatrix}
1 & 4 & -8 & -7
\end{bmatrix} = \begin{bmatrix}
-8 & -32 & 64 & 56
\end{bmatrix}
\]
2. **Adding the Scaled First Row to the Second Row (\(R_2\))**:
\[
R_2 = R_2 + (-8)R_1
\]
\[
\begin{bmatrix}
8 & 4 & 7 & 8
\end{bmatrix} + \begin{bmatrix}
-8 & -32 & 64 & 56
\end{bmatrix} = \begin{bmatrix}
0 & -28 & 71 & 64
\end{bmatrix}
\]
- For the 1st element: \(8 + (-8) = 0\)
- For the 2nd element: \(4 + (-32) = -28\)
- For the 3rd element: \(7 + 64 = 71\)
- For the 4th element: \(8 + 56 = 64\)
3. **Form the New Matrix with the Updated Second Row**
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