Compute the determinant of the matrix A, below, by using row operations to transform A to an upper-triangular matrix B. Then express the determinant of A as a multiple k of the determinant of B, and use this to compute the determinant of A. A = 7 3 -10 10 0 -10 10 -1 0 2017-1 7 3 -19 29 000 B- 0 0 0 = 000 det(A) k-det(B) = 0.0 = 0 One possible correct answer is: [7 3 10 10 10 -1 3 -3 00 0 10 B = 0-10 0 0 det(A) = k-det(B) = 1.-2100 = -2100
Compute the determinant of the matrix A, below, by using row operations to transform A to an upper-triangular matrix B. Then express the determinant of A as a multiple k of the determinant of B, and use this to compute the determinant of A. A = 7 3 -10 10 0 -10 10 -1 0 2017-1 7 3 -19 29 000 B- 0 0 0 = 000 det(A) k-det(B) = 0.0 = 0 One possible correct answer is: [7 3 10 10 10 -1 3 -3 00 0 10 B = 0-10 0 0 det(A) = k-det(B) = 1.-2100 = -2100
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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Please explain the answers with as much detail as possible. For the most part, I understand how the matrix was reduced. However, the confusion comes from the mathematical parts involving the multiplication and combining of each of the 4 created matrices as a result of the reduced 4X4 matrix. Thanks in advance!
![**Title: Determinant Computation Using Row Operations**
**Objective:**
Compute the determinant of the matrix \( A \) by transforming it into an upper-triangular matrix \( B \) using row operations. Then, express the determinant of \( A \) as a multiple \( k \) of the determinant of \( B \).
**Matrix \( A \):**
\[
A = \begin{bmatrix}
7 & 3 & -10 & 10 \\
0 & -10 & 10 & -1 \\
0 & 20 & -17 & -1 \\
7 & 3 & -19 & 29
\end{bmatrix}
\]
**Step 1: Initial Transformation (Incorrect Path):**
\[
B = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
- **Determinant Calculation:**
\[
\det(A) = k \cdot \det(B)
\]
\[
\det(A) = 0 \cdot 0
\]
\[
= 0
\]
**Step 2: Correct Transformation (Upper-Triangular Matrix):**
Transform \( A \) to matrix \( B \) using the correct series of row operations:
\[
B = \begin{bmatrix}
7 & 3 & -10 & 10 \\
0 & -10 & 10 & -1 \\
0 & 0 & 3 & -3 \\
0 & 0 & 0 & 10
\end{bmatrix}
\]
**Determinant Calculation:**
- The determinant of \( B \) is the product of its diagonal entries:
\[
\det(B) = 1 \cdot (-2100)
\]
\[
= -2100
\]
- Thus, the determinant of matrix \( A \) is:
\[
\det(A) = -2100
\]
This explanation includes the transformation of matrix \( A \) into its correct upper-triangular form \( B \), the calculation of \( \det(B) \), and subsequently the determinant of \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3eb3c55-3996-4e07-93a0-a3410210cd33%2F2668330c-1de8-47ec-b733-bd0194523bb0%2Fk97rc0n_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Determinant Computation Using Row Operations**
**Objective:**
Compute the determinant of the matrix \( A \) by transforming it into an upper-triangular matrix \( B \) using row operations. Then, express the determinant of \( A \) as a multiple \( k \) of the determinant of \( B \).
**Matrix \( A \):**
\[
A = \begin{bmatrix}
7 & 3 & -10 & 10 \\
0 & -10 & 10 & -1 \\
0 & 20 & -17 & -1 \\
7 & 3 & -19 & 29
\end{bmatrix}
\]
**Step 1: Initial Transformation (Incorrect Path):**
\[
B = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
- **Determinant Calculation:**
\[
\det(A) = k \cdot \det(B)
\]
\[
\det(A) = 0 \cdot 0
\]
\[
= 0
\]
**Step 2: Correct Transformation (Upper-Triangular Matrix):**
Transform \( A \) to matrix \( B \) using the correct series of row operations:
\[
B = \begin{bmatrix}
7 & 3 & -10 & 10 \\
0 & -10 & 10 & -1 \\
0 & 0 & 3 & -3 \\
0 & 0 & 0 & 10
\end{bmatrix}
\]
**Determinant Calculation:**
- The determinant of \( B \) is the product of its diagonal entries:
\[
\det(B) = 1 \cdot (-2100)
\]
\[
= -2100
\]
- Thus, the determinant of matrix \( A \) is:
\[
\det(A) = -2100
\]
This explanation includes the transformation of matrix \( A \) into its correct upper-triangular form \( B \), the calculation of \( \det(B) \), and subsequently the determinant of \( A \).
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