Discussion
The quadratic form corresponding to the matrix
\[ \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{bmatrix} \]
is :
Answer: A
The form is positive definite.
Use Sylvester's criterion on the leading principal minors:
\[ \begin{aligned} \Delta_{1} &= 2 \gt 0 \\ \Delta_{2} &= 2 \cdot 2 - 1 \cdot 1 = 3 \gt 0 \\ \Delta_{3} &= 2(2 - 1) - 1(1 - 0) + 0 = 1 \gt 0 \end{aligned} \]
All three are positive, so the quadratic form \(2x^{2} + 2y^{2} + z^{2} + 2xy + 2yz\) is positive for every non-zero vector.
A negative definite form needs minors alternating in sign starting negative, and a semidefinite form needs a zero determinant; neither fits a matrix whose determinant is 1 and whose first minor is 2.
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