Prove that $\mathsf{A}\mathsf{G}^\mathrm{T}\mathsf{G}$ is a positive semidefinite matrix

Question

Prove that $\mathsf{A}\mathsf{G}^\mathrm{T}\mathsf{G}$ is a positive semidefinite matrix, where $\mathsf{G} \in \mathbb{R}^{N \times N}$ and $\mathsf{A} = \mathrm{diag}(0,1,\dots,1,0) \in \mathbb{R}^{N \times N}$.

Answer 1

I think this is wrong. Take $G$ be a $3 \times 3$ matrix with all elements $1$. and take $x$ be the following matrix: $$ \begin{bmatrix} 1 \\ 1 \\ -4 \\ \end{bmatrix} $$ Then I think $x^\mathrm{T}\mathsf{A}\mathsf{G}^\mathrm{T}\mathsf{G}x$ is less than zero.