22. 4x² + 12x₁x₂ + 13x² = 16 23. - 11x + 84x₁x2 + 24x² = 156
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

Transcribed Image Text:22. 4x² + 12x₁x₂ + 13x² = 16
23. 11x + 84x₁x₂ + 24x²/
= 156
![25. Definiteness. A necessary and sufficient condition for
positive definiteness of a quadratic form Q(x) = x¹Ax
with symmetric matrix A is that all the principal minors
are positive (see Ref. [B3], vol. 1, p. 306), that is,
|a11
a11
a12
911 > 0,
a12
a12
a13
922 a23 > 0,
a12
a22
> 0,
det A > 0.
913
a23
a33
Show that the form in Prob. 22 is positive definite,
whereas that in Prob. 23 is indefinite.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F1e8f9cdf-18c7-421a-a93c-e6c123c6d365%2Ftlg1rq_processed.png&w=3840&q=75)
Transcribed Image Text:25. Definiteness. A necessary and sufficient condition for
positive definiteness of a quadratic form Q(x) = x¹Ax
with symmetric matrix A is that all the principal minors
are positive (see Ref. [B3], vol. 1, p. 306), that is,
|a11
a11
a12
911 > 0,
a12
a12
a13
922 a23 > 0,
a12
a22
> 0,
det A > 0.
913
a23
a33
Show that the form in Prob. 22 is positive definite,
whereas that in Prob. 23 is indefinite.
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