Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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](https://content.bartleby.com/qna-images/question/3e1fa912-5e0c-4d6e-bcb6-bc3942e62084/78b489b9-b0ec-46ce-bdbd-5da7cfda98d2/g3cs16e_processed.jpeg)
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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