Discussion
The quadratic form
\[ 4X_{1}^{2}+X_{2}^{2}+X_{3}^{2}-2X_{1}X_{2}+2X_{1}X_{3} \]
is :
Answer: C
The quadratic form is positive definite.
Its symmetric matrix and leading principal minors are
\[ \begin{aligned} &\begin{bmatrix} 4 & -1 & 1 \\ -1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} \\ &\Delta_{1} = 4,\ \Delta_{2} = 4 - 1 = 3 \\ &\Delta_{3} = 4(1) + 1(-1) + 1(-1) = 2 \end{aligned} \]
All three are positive, so by Sylvester's criterion the form is positive definite. Each off-diagonal entry is half the coefficient of the corresponding cross term.
A semidefinite form would need \(\Delta_{3} = 0\), and the negative options need \(\Delta_{1} \lt 0\).
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