Find the values of the unknown quantities for which each matrix is positive definite. 1 a A a 2 В 4 b |c C = d]
Find the values of the unknown quantities for which each matrix is positive definite. 1 a A a 2 В 4 b |c C = d]
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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Question
![**Title: Understanding Positive Definite Matrices**
**Objective:**
Determine the values of unknown quantities that ensure each given matrix is positive definite.
---
**Matrices:**
1. **Matrix A:**
\[
A = \begin{bmatrix}
1 & a \\
a & 9
\end{bmatrix}
\]
2. **Matrix B:**
\[
B = \begin{bmatrix}
2 & 4 \\
4 & b
\end{bmatrix}
\]
3. **Matrix C:**
\[
C = \begin{bmatrix}
c & d \\
d & c
\end{bmatrix}
\]
4. **Matrix D:**
\[
D = \begin{bmatrix}
d & 3 & 0 \\
3 & d & 4 \\
0 & 4 & d
\end{bmatrix}
\]
5. **Matrix S:**
\[
S = \begin{bmatrix}
s & -4 & -4 \\
-4 & s & -4 \\
-4 & -4 & s
\end{bmatrix}
\]
**Concept Review:**
A matrix is positive definite if it's symmetric and all its leading principal minors are positive. This concept is helpful in various fields such as differential equations, numerical analysis, and optimization.
**Steps to Determine Positive Definiteness:**
To ensure each matrix is positive definite:
- Confirm the matrix is symmetric.
- Calculate determinants of leading principal minors and ensure they are positive.
- Solve inequalities formed by these conditions to find the unknown values.
**Analysis and Solution:**
- For **Matrix A**, ensure \(1 > 0\) and the determinant \(1 \cdot 9 - a^2 > 0\).
- For **Matrix B**, ensure \(2 > 0\) and the determinant \(2b - 16 > 0\).
- For **Matrix C**, ensure \(c > 0\) and the determinant \(c^2 - d^2 > 0\).
- For **Matrix D**, analyze the determinants of \(1 \times 1\), \(2 \times 2\), and \(3 \times 3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7297a90f-0927-4369-b782-4368eb3beb88%2F0dea72e2-65a7-4e70-bcb0-496129f6a6c6%2Fbyez6gy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Positive Definite Matrices**
**Objective:**
Determine the values of unknown quantities that ensure each given matrix is positive definite.
---
**Matrices:**
1. **Matrix A:**
\[
A = \begin{bmatrix}
1 & a \\
a & 9
\end{bmatrix}
\]
2. **Matrix B:**
\[
B = \begin{bmatrix}
2 & 4 \\
4 & b
\end{bmatrix}
\]
3. **Matrix C:**
\[
C = \begin{bmatrix}
c & d \\
d & c
\end{bmatrix}
\]
4. **Matrix D:**
\[
D = \begin{bmatrix}
d & 3 & 0 \\
3 & d & 4 \\
0 & 4 & d
\end{bmatrix}
\]
5. **Matrix S:**
\[
S = \begin{bmatrix}
s & -4 & -4 \\
-4 & s & -4 \\
-4 & -4 & s
\end{bmatrix}
\]
**Concept Review:**
A matrix is positive definite if it's symmetric and all its leading principal minors are positive. This concept is helpful in various fields such as differential equations, numerical analysis, and optimization.
**Steps to Determine Positive Definiteness:**
To ensure each matrix is positive definite:
- Confirm the matrix is symmetric.
- Calculate determinants of leading principal minors and ensure they are positive.
- Solve inequalities formed by these conditions to find the unknown values.
**Analysis and Solution:**
- For **Matrix A**, ensure \(1 > 0\) and the determinant \(1 \cdot 9 - a^2 > 0\).
- For **Matrix B**, ensure \(2 > 0\) and the determinant \(2b - 16 > 0\).
- For **Matrix C**, ensure \(c > 0\) and the determinant \(c^2 - d^2 > 0\).
- For **Matrix D**, analyze the determinants of \(1 \times 1\), \(2 \times 2\), and \(3 \times 3
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