Find the condition on b such that the matrix given below, is positive definite. A = 2 -1 b 2 -1 2 b -1
Find the condition on b such that the matrix given below, is positive definite. A = 2 -1 b 2 -1 2 b -1
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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matrix analysis practice question,Please write the simplest answer process,thanks
![### Linear Algebra: Positive Definite and Orthogonal Matrices
**Question 1:**
**(a)** Find the condition on \( b \) such that the matrix given below is **positive definite**.
\[
A = \begin{pmatrix}
2 & -1 & b \\
-1 & 2 & -1 \\
b & -1 & 2
\end{pmatrix}
\]
**(b)** Find the values of \( \alpha \), \( \beta \) such that the following matrix is an **orthogonal matrix**.
\[
B = \frac{1}{3} \begin{pmatrix}
2 & -2 & \alpha \\
2 & \beta & -2 \\
\alpha & 2 & 2
\end{pmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F979d0aba-5428-414f-a3ba-5510f0301082%2Ffe4f8df7-e9fb-492c-a2cb-82a12837cae2%2F1w421a7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Positive Definite and Orthogonal Matrices
**Question 1:**
**(a)** Find the condition on \( b \) such that the matrix given below is **positive definite**.
\[
A = \begin{pmatrix}
2 & -1 & b \\
-1 & 2 & -1 \\
b & -1 & 2
\end{pmatrix}
\]
**(b)** Find the values of \( \alpha \), \( \beta \) such that the following matrix is an **orthogonal matrix**.
\[
B = \frac{1}{3} \begin{pmatrix}
2 & -2 & \alpha \\
2 & \beta & -2 \\
\alpha & 2 & 2
\end{pmatrix}
\]
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