Use expansion by cofactors to find the determinant of the matrix. W y 24 -15 21 18 -32 10 -24 18 -32 35 22 -40
Use expansion by cofactors to find the determinant of the matrix. W y 24 -15 21 18 -32 10 -24 18 -32 35 22 -40
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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Linear Algrebra
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![**Matrix Determinant Calculation Using Cofactor Expansion**
To find the determinant of the following matrix using expansion by cofactors, follow the steps provided. Here's the given matrix:
\[
\begin{bmatrix}
w & x & y & z \\
24 & -15 & 21 & 18 \\
-32 & 10 & -24 & 18 \\
-32 & 35 & 22 & -40 \\
\end{bmatrix}
\]
**Matrix Description:**
- The matrix is a 4x4 matrix.
- The first row consists of variables \( w, x, y, z \).
- The remaining rows contain the following numerical values:
- Second row: 24, -15, 21, 18
- Third row: -32, 10, -24, 18
- Fourth row: -32, 35, 22, -40
**Instructions for Finding the Determinant:**
1. Select one row or column for expansion. Typically, choosing a row or column with zeros simplifies the calculation, though none are present here.
2. Calculate the cofactor for each element in the selected row or column. The cofactor involves computing the determinant of the smaller 3x3 matrix that remains when the row and column of the element are removed.
3. Apply the cofactor expansion formula, summing the products of the matrix element and its cofactor, considering the sign (+/-) based on the position in the matrix.
**Example Calculation Step (for educational purposes only):**
Choose the first row for the expansion:
- \( \text{Det}(Matrix) = w \cdot \text{Cofactor}(w) - x \cdot \text{Cofactor}(x) + y \cdot \text{Cofactor}(y) - z \cdot \text{Cofactor}(z) \)
**Note: The actual computation requires solving each smaller 3x3 determinant, which can be detailed on expanding each element based on their minors and cofactors.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc42b80bf-a5d4-414b-bce1-0fe52a04dbbd%2Ff63478f5-710f-4558-8957-13bb0df9f616%2Fa0bsrdg_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Determinant Calculation Using Cofactor Expansion**
To find the determinant of the following matrix using expansion by cofactors, follow the steps provided. Here's the given matrix:
\[
\begin{bmatrix}
w & x & y & z \\
24 & -15 & 21 & 18 \\
-32 & 10 & -24 & 18 \\
-32 & 35 & 22 & -40 \\
\end{bmatrix}
\]
**Matrix Description:**
- The matrix is a 4x4 matrix.
- The first row consists of variables \( w, x, y, z \).
- The remaining rows contain the following numerical values:
- Second row: 24, -15, 21, 18
- Third row: -32, 10, -24, 18
- Fourth row: -32, 35, 22, -40
**Instructions for Finding the Determinant:**
1. Select one row or column for expansion. Typically, choosing a row or column with zeros simplifies the calculation, though none are present here.
2. Calculate the cofactor for each element in the selected row or column. The cofactor involves computing the determinant of the smaller 3x3 matrix that remains when the row and column of the element are removed.
3. Apply the cofactor expansion formula, summing the products of the matrix element and its cofactor, considering the sign (+/-) based on the position in the matrix.
**Example Calculation Step (for educational purposes only):**
Choose the first row for the expansion:
- \( \text{Det}(Matrix) = w \cdot \text{Cofactor}(w) - x \cdot \text{Cofactor}(x) + y \cdot \text{Cofactor}(y) - z \cdot \text{Cofactor}(z) \)
**Note: The actual computation requires solving each smaller 3x3 determinant, which can be detailed on expanding each element based on their minors and cofactors.**
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