$A^2$ is a positive definite matrix.

Question

Let $A$ be an $n$x$n$ non-singular symmetric matrix with postive entries. Then which of the following are correct? $1)$ $A$ is positive definite. $2)$ $A^{-1}$ is a matrix with postive entries. $3)$ $A^2$ is positive definite.

First two are wrong, since the matrix $A=\begin{pmatrix}1&2\\2&1\end{pmatrix}$ is of determinant $-3$. Then it fails above two. Please help me to prove the last one.

Answer 1

For any vector $v$, $$v^\top A^2 v = \|A v\|^2 \ge 0$$