4. Compute the determinant using the Laplace expansion (using the first now and the Sonus nule for: A= 135 246 024J
4. Compute the determinant using the Laplace expansion (using the first now and the Sonus nule for: A= 135 246 024J
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
4.1
![**Title: Calculating the Determinant of a Matrix using Laplace Expansion**
**Objective:** Learn how to compute the determinant of a matrix using the Laplace expansion method, specifically focusing on the first row.
**Example:**
We have the following 3x3 matrix \( A \):
\[
A = \begin{bmatrix}
1 & 3 & 5 \\
2 & 4 & 6 \\
0 & 2 & 4 \\
\end{bmatrix}
\]
**Instructions:**
1. **Select a Row or Column:** Begin by selecting a row or column for the expansion. In this example, we will use the first row.
2. **Apply the Laplace Expansion Formula:** The determinant of matrix \( A \), denoted as \( \text{det}(A) \), can be calculated using the formula:
\[
\text{det}(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \cdot \text{det}(A_{ij})
\]
Where \( a_{ij} \) is the element in the \( i \)-th row and \( j \)-th column, and \( A_{ij} \) is the submatrix formed by deleting the \( i \)-th row and \( j \)-th column.
3. **Calculate Each Cofactor:** For the selected row (first row):
- For element \( a_{11} = 1 \):
\[
\text{Cofactor} = (-1)^{1+1} \cdot 1 \cdot \text{det}\left(\begin{bmatrix}
4 & 6 \\
2 & 4 \\
\end{bmatrix}\right)
\]
- For element \( a_{12} = 3 \):
\[
\text{Cofactor} = (-1)^{1+2} \cdot 3 \cdot \text{det}\left(\begin{bmatrix}
2 & 6 \\
0 & 4 \\
\end{bmatrix}\right)
\]
- For element \( a_{13} = 5 \):
\[
\text{Cofactor} = (-1)^{1+3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e25041d-7573-46df-b9d3-ec2dd7694c16%2F481a0d5b-1a62-404a-a8e5-a379fe175fec%2Ffu0kaec_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Determinant of a Matrix using Laplace Expansion**
**Objective:** Learn how to compute the determinant of a matrix using the Laplace expansion method, specifically focusing on the first row.
**Example:**
We have the following 3x3 matrix \( A \):
\[
A = \begin{bmatrix}
1 & 3 & 5 \\
2 & 4 & 6 \\
0 & 2 & 4 \\
\end{bmatrix}
\]
**Instructions:**
1. **Select a Row or Column:** Begin by selecting a row or column for the expansion. In this example, we will use the first row.
2. **Apply the Laplace Expansion Formula:** The determinant of matrix \( A \), denoted as \( \text{det}(A) \), can be calculated using the formula:
\[
\text{det}(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \cdot \text{det}(A_{ij})
\]
Where \( a_{ij} \) is the element in the \( i \)-th row and \( j \)-th column, and \( A_{ij} \) is the submatrix formed by deleting the \( i \)-th row and \( j \)-th column.
3. **Calculate Each Cofactor:** For the selected row (first row):
- For element \( a_{11} = 1 \):
\[
\text{Cofactor} = (-1)^{1+1} \cdot 1 \cdot \text{det}\left(\begin{bmatrix}
4 & 6 \\
2 & 4 \\
\end{bmatrix}\right)
\]
- For element \( a_{12} = 3 \):
\[
\text{Cofactor} = (-1)^{1+2} \cdot 3 \cdot \text{det}\left(\begin{bmatrix}
2 & 6 \\
0 & 4 \\
\end{bmatrix}\right)
\]
- For element \( a_{13} = 5 \):
\[
\text{Cofactor} = (-1)^{1+3
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