# Sufficient conditions for Loewner ordering of two matrices?

Le $$\mathbf{A}$$ and $$\mathbf{B}$$ two psd matrices of size $$n$$. Additionally we assume that the entries are real and non-negative. Does the following hold: $$\forall (i,j) \in [n], \mathbf{A}_{ij} \leq \mathbf{B}_{ij} \implies \mathbf{A}^2 \preccurlyeq \mathbf{B}^2$$ where $$\preccurlyeq$$ is the Loewner partial order on the cone of psd matrices.

No. E.g. $$A=\pmatrix{1&0\\ 0&0}\le \pmatrix{1&1\\ 1&1}=B$$ entrywise, but $$B^2-A^2=\pmatrix{1&2\\ 2&2}$$ is indefinite.