linear algebra - further properties of positive definite matrix

Question

Tried to prove/disprove the following statements:

1. [$A, B \in \mathbb{M}_{n \times n}(\mathbb{R})$ are positive definite] $\implies$ (AB is positive definite )

2. [all eigenvalues $\lambda$ of $A \in \mathbb{M}_{n \times n}(\mathbb{R}) $ are $\lambda > 0$] $\implies $ ($A$ is positive definite)

Thought about finding counterexamples, but my every attempt failed.

Any help/advice/couterexample/proof very, very appreciated!

Answer 1

2 is clearly false

1 -5 0 1 has e. Values 1 but this is not clearly positive definite

However if you further assume the matrix is symmetric then the result is true

Answer 2

1. AB needs to be symmetric in order to be positive definite. But a product of symmetric matrices is symmetric iff they commute, so it's easy to construct a counterexample: $\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}\begin{pmatrix}2 & 1\\1 & 2\end{pmatrix}=\begin{pmatrix}2 & 1\\2 & 4\end{pmatrix}$.
2. You can take a positive jordan block of dimension>1. If you further assume that A is symmetric, then A is orthogonally diagonalizable and you can use this to prove that A is positive definite.