Discussion
Corresponding to the three variables X1, X2, X3 the quadrate form \(X_{1}^{2} + 2X_{2}^{2} + 4X_{3}^{2} - X_{1}X_{2} + X_{1}X_{3}\) is :
Answer: A
The quadratic form is positive definite.
Its symmetric matrix is
\[ A = \begin{pmatrix} 1 & -\frac{1}{2} & \frac{1}{2} \\ -\frac{1}{2} & 2 & 0 \\ \frac{1}{2} & 0 & 4 \end{pmatrix} \]
The leading principal minors are 1, \(2 - \frac{1}{4} = \frac{7}{4}\), and
\[ \det A = 8 - 1 - \tfrac{1}{2} = \tfrac{13}{2} \]
All three are positive, so by Sylvester's criterion the form is positive definite. Positive semidefinite would allow a zero minor, and the negative cases need the minors to alternate in sign.
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