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

Elementary Linear Algebra (MindTap Course List)
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Chapter4: Vector Spaces
Section4.6: Rank Of A Matrix And Systems Of Linear Equations
Problem 77E: Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and...
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22. 4x² + 12x₁x₂ + 13x² = 16
23. 11x + 84x₁x₂ + 24x²/
= 156
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.
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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