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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![**Problem Statement:**
Find the values of \( k \), if any, that satisfy the equation:
\[
\begin{bmatrix}
k & 1 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -3
\end{bmatrix}
\begin{bmatrix}
k \\
1 \\
1
\end{bmatrix}
= 0
\]
**Explanation:**
The problem requires finding values of \( k \) that make the above equation true. The equation is an expression involving the multiplication of matrices and vectors.
1. **Matrix and Vector Notations:**
- The first part \(\begin{bmatrix} k & 1 & 1 \end{bmatrix}\) is a 1x3 matrix (or row vector).
- The second part is a 3x3 matrix:
\[
\begin{bmatrix}
1 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -3
\end{bmatrix}
\]
- The third part \(\begin{bmatrix} k \\ 1 \\ 1 \end{bmatrix}\) is a 3x1 matrix (or column vector).
2. **Matrix Multiplication:**
- First, multiply the 3x3 matrix by the 3x1 column vector.
- Then, multiply the resulting 1x3 row vector by the 1x3 matrix on the left.
3. **Objective:**
- Solve the resulting expression for \( k \) that results in the overall equation equaling zero. This requires knowledge of matrix algebra and solving resulting polynomial or linear equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86847a45-621f-4eb2-b143-0cde4fac410f%2Ffa1ee814-ddbb-4429-bf25-e5760b7c4fda%2F1m9rzdh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the values of \( k \), if any, that satisfy the equation:
\[
\begin{bmatrix}
k & 1 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -3
\end{bmatrix}
\begin{bmatrix}
k \\
1 \\
1
\end{bmatrix}
= 0
\]
**Explanation:**
The problem requires finding values of \( k \) that make the above equation true. The equation is an expression involving the multiplication of matrices and vectors.
1. **Matrix and Vector Notations:**
- The first part \(\begin{bmatrix} k & 1 & 1 \end{bmatrix}\) is a 1x3 matrix (or row vector).
- The second part is a 3x3 matrix:
\[
\begin{bmatrix}
1 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -3
\end{bmatrix}
\]
- The third part \(\begin{bmatrix} k \\ 1 \\ 1 \end{bmatrix}\) is a 3x1 matrix (or column vector).
2. **Matrix Multiplication:**
- First, multiply the 3x3 matrix by the 3x1 column vector.
- Then, multiply the resulting 1x3 row vector by the 1x3 matrix on the left.
3. **Objective:**
- Solve the resulting expression for \( k \) that results in the overall equation equaling zero. This requires knowledge of matrix algebra and solving resulting polynomial or linear equations.
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