Quadratic forms & definiteness

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   The quadratic form $Q(x, y) = -3x^{2} + 6xy + 3y^{2}$ has symmetric matrix $A = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}$. When $\det(A) \ne 0$, the sign of $\det(A)$ together with the sign of the leading entry $-3$ tells whether $Q$ is positive definite, negative definite, or indefinite. Compute $\det(A)$.

2. Warm-up

   The quadratic form $Q(x, y) = -4x^{2} + 8xy - y^{2}$ has symmetric matrix $A = \begin{pmatrix} -4 & 4 \\ 4 & -1 \end{pmatrix}$. When $\det(A) \ne 0$, the sign of $\det(A)$ together with the sign of the leading entry $-4$ tells whether $Q$ is positive definite, negative definite, or indefinite. Compute $\det(A)$.

3. Warm-up

   The quadratic form $Q(x, y) = x^{2} + 8xy + 4y^{2}$ has symmetric matrix $A = \begin{pmatrix} 1 & 4 \\ 4 & 4 \end{pmatrix}$. When $\det(A) \ne 0$, the sign of $\det(A)$ together with the sign of the leading entry $1$ tells whether $Q$ is positive definite, negative definite, or indefinite. Compute $\det(A)$.

Problems shown under their original terms: Generated (template, solver-computed).