Compute the following determinant by cofactor expansions. At each step, choose the row or column that involves the least amount of 1-2 42 0 0 20 5 -4 -5 3 30 35 The determinant is ...
Compute the following determinant by cofactor expansions. At each step, choose the row or column that involves the least amount of 1-2 42 0 0 20 5 -4 -5 3 30 35 The determinant is ...
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
![### Determinant Calculation Using Cofactor Expansions
**Exercise:**
Compute the following determinant by cofactor expansions. At each step, choose the row or column that involves the least amount of computation.
\[
\begin{vmatrix}
1 & -2 & 4 & 2 \\
0 & 0 & 2 & 0 \\
5 & -4 & -5 & 3 \\
3 & 0 & 3 & 5
\end{vmatrix}
\]
---
### Explanation:
1. **Matrix Provided:**
\[
\begin{bmatrix}
1 & -2 & 4 & 2 \\
0 & 0 & 2 & 0 \\
5 & -4 & -5 & 3 \\
3 & 0 & 3 & 5
\end{bmatrix}
\]
2. **Steps:**
- Identify the row or column with the most zeros to minimize computation.
- Apply cofactor expansion on the selected row or column.
**Note:** Here, the second row has the most zeros \( [0, 0, 2, 0] \), which simplifies the computation significantly.
3. **Expansion Process:**
Expanding along the second row:
- The second row is \( [0, 0, 2, 0] \)
- Only the third element (2) is non-zero.
- The corresponding minor matrix for this element is obtained by deleting the row and column containing this element.
\[
\begin{vmatrix}
1 & -2 & 2 \\
5 & -4 & 3 \\
3 & 0 & 5
\end{vmatrix}
\]
4. **Continue with 3x3 Determinant:**
Compute the determinant of the above 3x3 matrix using cofactor expansion on the row or column with the least non-zero elements for simplification.
---
### Conclusion:
After following the necessary steps of cofactor expansion, the determinant can be computed.
**Answer:**
The determinant is [ ____ ].
---
### Graphs or Diagrams:
- There are no additional graphs or diagrams in the provided content.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a1a8c69-ac20-486b-9f67-8f66504c5494%2F5af39b28-96b8-4392-9c6f-639cbbeaab23%2Fc37t27_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determinant Calculation Using Cofactor Expansions
**Exercise:**
Compute the following determinant by cofactor expansions. At each step, choose the row or column that involves the least amount of computation.
\[
\begin{vmatrix}
1 & -2 & 4 & 2 \\
0 & 0 & 2 & 0 \\
5 & -4 & -5 & 3 \\
3 & 0 & 3 & 5
\end{vmatrix}
\]
---
### Explanation:
1. **Matrix Provided:**
\[
\begin{bmatrix}
1 & -2 & 4 & 2 \\
0 & 0 & 2 & 0 \\
5 & -4 & -5 & 3 \\
3 & 0 & 3 & 5
\end{bmatrix}
\]
2. **Steps:**
- Identify the row or column with the most zeros to minimize computation.
- Apply cofactor expansion on the selected row or column.
**Note:** Here, the second row has the most zeros \( [0, 0, 2, 0] \), which simplifies the computation significantly.
3. **Expansion Process:**
Expanding along the second row:
- The second row is \( [0, 0, 2, 0] \)
- Only the third element (2) is non-zero.
- The corresponding minor matrix for this element is obtained by deleting the row and column containing this element.
\[
\begin{vmatrix}
1 & -2 & 2 \\
5 & -4 & 3 \\
3 & 0 & 5
\end{vmatrix}
\]
4. **Continue with 3x3 Determinant:**
Compute the determinant of the above 3x3 matrix using cofactor expansion on the row or column with the least non-zero elements for simplification.
---
### Conclusion:
After following the necessary steps of cofactor expansion, the determinant can be computed.
**Answer:**
The determinant is [ ____ ].
---
### Graphs or Diagrams:
- There are no additional graphs or diagrams in the provided content.
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