Find the values of k, if any, that satisfy the equation. 1 k [k 1 1] 1 1=0 0 2 -3 1

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.
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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