Use expansion by cofactors to find the determinant of the matrix. 46007 0 1 2 3 3 0 0 24 1 0 0 5 3 6 0 0 0 0 3 X -54
Use expansion by cofactors to find the determinant of the matrix. 46007 0 1 2 3 3 0 0 24 1 0 0 5 3 6 0 0 0 0 3 X -54
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 Calculation by Expansion of Cofactors
#### Problem Statement:
Use expansion by cofactors to find the determinant of the matrix.
#### Matrix:
\[
\begin{bmatrix}
4 & 6 & 0 & 0 & 7 \\
0 & 1 & 2 & 3 & 3 \\
0 & 0 & 2 & 4 & 1 \\
0 & 0 & 5 & 3 & 6 \\
0 & 0 & 0 & 0 & 3
\end{bmatrix}
\]
#### Submitted Answer:
-54 ❌
#### Explanation:
The matrix shown is a 5x5 matrix, composed of integers. The task is to find the determinant using the method of cofactor expansion. The originally submitted determinant of -54 is incorrect, as indicated by the ❌ symbol.
#### Steps for Solution:
1. **Choosing a Row/Column:**
- Usually, rows or columns with the most zeros are selected to simplify calculations. In this case, Row 1 or Column 1 could be good options.
2. **Calculating the Determinant:**
- Use the expansion formula:
\[
\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + \cdots + a_{1n}C_{1n}
\]
- \( a_{ij} \) are the elements of the chosen row/column.
- \( C_{ij} \) are the cofactors corresponding to each element.
- The cofactor \( C_{ij} = (-1)^{i+j} M_{ij} \), where \( M_{ij} \) is the minor determinant after removing the i-th row and j-th column.
3. **Final Calculation:**
- Execute the calculations for minor determinants and apply them with the respective matrix elements.
- Sum all computed values to find the determinant.
#### Future Tips:
- Double-check your calculations, especially the signs of cofactors and minor determinants.
- Make use of properties of matrices to recognize patterns that can simplify the determinant calculation.
This methodology is crucial for solving higher-dimensional matrix determinants where manual computation is involved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8eeaacd3-f9c7-4223-a8e6-a717e551a98a%2Fddec658d-139a-4161-b44e-e059276a29fa%2Fhltepfd_processed.png&w=3840&q=75)
Transcribed Image Text:### Determinant Calculation by Expansion of Cofactors
#### Problem Statement:
Use expansion by cofactors to find the determinant of the matrix.
#### Matrix:
\[
\begin{bmatrix}
4 & 6 & 0 & 0 & 7 \\
0 & 1 & 2 & 3 & 3 \\
0 & 0 & 2 & 4 & 1 \\
0 & 0 & 5 & 3 & 6 \\
0 & 0 & 0 & 0 & 3
\end{bmatrix}
\]
#### Submitted Answer:
-54 ❌
#### Explanation:
The matrix shown is a 5x5 matrix, composed of integers. The task is to find the determinant using the method of cofactor expansion. The originally submitted determinant of -54 is incorrect, as indicated by the ❌ symbol.
#### Steps for Solution:
1. **Choosing a Row/Column:**
- Usually, rows or columns with the most zeros are selected to simplify calculations. In this case, Row 1 or Column 1 could be good options.
2. **Calculating the Determinant:**
- Use the expansion formula:
\[
\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + \cdots + a_{1n}C_{1n}
\]
- \( a_{ij} \) are the elements of the chosen row/column.
- \( C_{ij} \) are the cofactors corresponding to each element.
- The cofactor \( C_{ij} = (-1)^{i+j} M_{ij} \), where \( M_{ij} \) is the minor determinant after removing the i-th row and j-th column.
3. **Final Calculation:**
- Execute the calculations for minor determinants and apply them with the respective matrix elements.
- Sum all computed values to find the determinant.
#### Future Tips:
- Double-check your calculations, especially the signs of cofactors and minor determinants.
- Make use of properties of matrices to recognize patterns that can simplify the determinant calculation.
This methodology is crucial for solving higher-dimensional matrix determinants where manual computation is involved.
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