A = -3 2 -1 1 k A = 1 k² k k k² 050 [1 k k k² 7 1 5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
In Parts A & b, evaluate det(A) by a cofactor expansion along a row or column of your choice.
![Below are two matrices labeled \( A \), each containing different elements.
The first matrix \( A \) is a 3x3 matrix with the following elements:
\[ A = \begin{pmatrix}
-3 & 0 & 7 \\
2 & 5 & 1 \\
-1 & 0 & 5
\end{pmatrix} \]
This matrix has entries ranging from \(-3\) to \(7\), organized in three rows and three columns.
The second matrix \( A \) is also a 3x3 matrix but with elements defined in terms of a variable \( k \):
\[ A = \begin{pmatrix}
1 & k & k^2 \\
1 & k & k^2 \\
1 & k & k^2
\end{pmatrix} \]
In this matrix, the elements in each row are identical, comprising \( 1 \), \( k \), and \( k^2 \). This structure results in three identical rows.
### Explanation:
1. **First Matrix**
- **Element in the first row, first column**: -3
- **Element in the first row, second column**: 0
- **Element in the first row, third column**: 7
- **Element in the second row, first column**: 2
- **Element in the second row, second column**: 5
- **Element in the second row, third column**: 1
- **Element in the third row, first column**: -1
- **Element in the third row, second column**: 0
- **Element in the third row, third column**: 5
2. **Second Matrix**
- **Each row contains the following elements**: 1, \( k \), and \( k^2 \)
These matrices are typically used to illustrate different concepts in linear algebra, such as matrix operations, determinants, and the properties of matrix elements.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87643561-f7f8-47c3-82b0-7735ae0706bf%2F65b706e0-4d61-4173-9794-7df6f3d0e3ab%2Fqlquiro_processed.png&w=3840&q=75)
Transcribed Image Text:Below are two matrices labeled \( A \), each containing different elements.
The first matrix \( A \) is a 3x3 matrix with the following elements:
\[ A = \begin{pmatrix}
-3 & 0 & 7 \\
2 & 5 & 1 \\
-1 & 0 & 5
\end{pmatrix} \]
This matrix has entries ranging from \(-3\) to \(7\), organized in three rows and three columns.
The second matrix \( A \) is also a 3x3 matrix but with elements defined in terms of a variable \( k \):
\[ A = \begin{pmatrix}
1 & k & k^2 \\
1 & k & k^2 \\
1 & k & k^2
\end{pmatrix} \]
In this matrix, the elements in each row are identical, comprising \( 1 \), \( k \), and \( k^2 \). This structure results in three identical rows.
### Explanation:
1. **First Matrix**
- **Element in the first row, first column**: -3
- **Element in the first row, second column**: 0
- **Element in the first row, third column**: 7
- **Element in the second row, first column**: 2
- **Element in the second row, second column**: 5
- **Element in the second row, third column**: 1
- **Element in the third row, first column**: -1
- **Element in the third row, second column**: 0
- **Element in the third row, third column**: 5
2. **Second Matrix**
- **Each row contains the following elements**: 1, \( k \), and \( k^2 \)
These matrices are typically used to illustrate different concepts in linear algebra, such as matrix operations, determinants, and the properties of matrix elements.
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