3. 2-2 3 1 1 3 3 2 -1
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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![### Cofactor Expansion for Determinants
Compute the determinants in Exercises 1–8 using a cofactor expansion across the first row. In Exercises 1–4, also compute the determinant by a cofactor expansion down the second column.
#### Exercise 1
\[
\begin{vmatrix}
3 & 0 & 4 \\
2 & 3 & 2 \\
0 & 5 & -1 \\
\end{vmatrix}
\]
#### Exercise 2
\[
\begin{vmatrix}
0 & 4 & 1 \\
5 & -3 & 0 \\
2 & 4 & 1 \\
\end{vmatrix}
\]
#### Exercise 3
\[
\begin{vmatrix}
2 & -2 & 3 \\
3 & 1 & 2 \\
1 & 3 & -1 \\
\end{vmatrix}
\]
To compute the determinant for these matrices using the cofactor expansion method:
1. **Cofactor Expansion Across the First Row**:
- Select the elements from the first row.
- Compute the minor for each element by removing the row and column of the current element.
- Multiply the element by its cofactor, which is the determinant of the minor matrix, adjusting by the sign according to the position (-1)^(i+j).
- Sum these values to get the determinant.
2. **Cofactor Expansion Down the Second Column**:
- Select the elements from the second column.
- Compute the minor for each element by removing the row and column of the current element.
- Multiply the element by its cofactor, which is the determinant of the minor matrix, adjusting by the sign according to the position (-1)^(i+j).
- Sum these values to get the determinant.
By breaking down matrices and understanding their cofactors, you'll gain a deeper insight into how determinants reveal properties of linear transformations and matrix equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2Fadcb85ee-8e67-4895-a2d2-2cd2515206d8%2Fspfrpaf_processed.png&w=3840&q=75)
Transcribed Image Text:### Cofactor Expansion for Determinants
Compute the determinants in Exercises 1–8 using a cofactor expansion across the first row. In Exercises 1–4, also compute the determinant by a cofactor expansion down the second column.
#### Exercise 1
\[
\begin{vmatrix}
3 & 0 & 4 \\
2 & 3 & 2 \\
0 & 5 & -1 \\
\end{vmatrix}
\]
#### Exercise 2
\[
\begin{vmatrix}
0 & 4 & 1 \\
5 & -3 & 0 \\
2 & 4 & 1 \\
\end{vmatrix}
\]
#### Exercise 3
\[
\begin{vmatrix}
2 & -2 & 3 \\
3 & 1 & 2 \\
1 & 3 & -1 \\
\end{vmatrix}
\]
To compute the determinant for these matrices using the cofactor expansion method:
1. **Cofactor Expansion Across the First Row**:
- Select the elements from the first row.
- Compute the minor for each element by removing the row and column of the current element.
- Multiply the element by its cofactor, which is the determinant of the minor matrix, adjusting by the sign according to the position (-1)^(i+j).
- Sum these values to get the determinant.
2. **Cofactor Expansion Down the Second Column**:
- Select the elements from the second column.
- Compute the minor for each element by removing the row and column of the current element.
- Multiply the element by its cofactor, which is the determinant of the minor matrix, adjusting by the sign according to the position (-1)^(i+j).
- Sum these values to get the determinant.
By breaking down matrices and understanding their cofactors, you'll gain a deeper insight into how determinants reveal properties of linear transformations and matrix equations.
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