Use expansion by cofactors to find the determinant of the matrix. 32003 0 1 5 3 2 00465 0024 5 0000 7
Use expansion by cofactors to find the determinant of the matrix. 32003 0 1 5 3 2 00465 0024 5 0000 7
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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![### Determinant of a Matrix Using Cofactor Expansion
To find the determinant of a given matrix using cofactor expansion, follow the detailed example below:
In this case, we will work with the following 5x5 matrix:
\[
\begin{bmatrix}
3 & 2 & 0 & 0 & 3 \\
0 & 1 & 5 & 3 & 2 \\
0 & 0 & 4 & 6 & 5 \\
0 & 0 & 2 & 4 & 5 \\
0 & 0 & 0 & 0 & 7
\end{bmatrix}
\]
### Step-by-step procedure:
1. **Identify Rows and Columns:** Choose any row or column to expand along. It is often helpful to pick a row or column with the most zeros to simplify calculations.
2. **Expand Along a Row or Column:** For this example, let's expand along the first row:
\[
\text{det}(A) = 3 \cdot \text{det}\begin{bmatrix}
1 & 5 & 3 & 2 \\
0 & 4 & 6 & 5 \\
0 & 2 & 4 & 5 \\
0 & 0 & 0 & 7
\end{bmatrix}
- 2 \cdot \text{det}\begin{bmatrix}
0 & 5 & 3 & 2 \\
0 & 4 & 6 & 5 \\
0 & 2 & 4 & 5 \\
0 & 0 & 0 & 7
\end{bmatrix}
+ 0 - 0 + 0 \cdot \text{det of appropriate sub-matrix}
\]
3. **Determinant of Sub-matrices:** Continue expanding determinants of resulting sub-matrices using the same method until reaching a 2x2 matrix, where the determinant can be easily calculated:
For example:
\[
\text{det}\begin{bmatrix}
1 & 5 & 3 & 2 \\
0 & 4 & 6 & 5 \\
0 & 2 & 4 & 5 \\
0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e1fa912-5e0c-4d6e-bcb6-bc3942e62084%2F78b489b9-b0ec-46ce-bdbd-5da7cfda98d2%2Fg3cs16e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determinant of a Matrix Using Cofactor Expansion
To find the determinant of a given matrix using cofactor expansion, follow the detailed example below:
In this case, we will work with the following 5x5 matrix:
\[
\begin{bmatrix}
3 & 2 & 0 & 0 & 3 \\
0 & 1 & 5 & 3 & 2 \\
0 & 0 & 4 & 6 & 5 \\
0 & 0 & 2 & 4 & 5 \\
0 & 0 & 0 & 0 & 7
\end{bmatrix}
\]
### Step-by-step procedure:
1. **Identify Rows and Columns:** Choose any row or column to expand along. It is often helpful to pick a row or column with the most zeros to simplify calculations.
2. **Expand Along a Row or Column:** For this example, let's expand along the first row:
\[
\text{det}(A) = 3 \cdot \text{det}\begin{bmatrix}
1 & 5 & 3 & 2 \\
0 & 4 & 6 & 5 \\
0 & 2 & 4 & 5 \\
0 & 0 & 0 & 7
\end{bmatrix}
- 2 \cdot \text{det}\begin{bmatrix}
0 & 5 & 3 & 2 \\
0 & 4 & 6 & 5 \\
0 & 2 & 4 & 5 \\
0 & 0 & 0 & 7
\end{bmatrix}
+ 0 - 0 + 0 \cdot \text{det of appropriate sub-matrix}
\]
3. **Determinant of Sub-matrices:** Continue expanding determinants of resulting sub-matrices using the same method until reaching a 2x2 matrix, where the determinant can be easily calculated:
For example:
\[
\text{det}\begin{bmatrix}
1 & 5 & 3 & 2 \\
0 & 4 & 6 & 5 \\
0 & 2 & 4 & 5 \\
0
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