Positive semidefinite and positive definite matrices

Determine by direct calculation (i.e., without calculating the characteristic polynomial or the eigenvalues) which of the following matrices are positive definite, positive semidefinite, or neither.

Calculate the eigenvalues of each symmetric matrix, then determine for each matrix whether it is positive definite, positive semidefinite, or neither.

is positive definite, positive semidefinite, or neither.

Which of the following formulas define an inner product on $\mathbb{R}^{3}$? Here, $\mathbf{u}=\left[u_{1},u_{2},u_{3}\right]^{T}$ and $\mathbf{v}=\left[v_{1},v_{2},v_{3}\right]^{T}$.

1. $\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}+2u_{1}v_{2}+2u_{2}v_{1}+3u_{2}v_{2}+3u_{3}v_{3}$.

2. $\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}-u_{1}v_{2}-u_{2}v_{1}+2u_{2}v_{2}+u_{3}v_{3}$.

3. $\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}-u_{1}v_{2}-u_{2}v_{1}+u_{2}v_{2}+4u_{3}v_{3}$.

4. $\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}-u_{1}v_{2}-u_{2}v_{1}+3u_{2}v_{2}-u_{2}v_{3}-u_{3}v_{2}+u_{3}v_{3}$.

Use Descartes’ rule of signs to determine which of the following matrices are positive definite and/or positive semidefinite.