I have a matrix $$A(x) = \frac{-1}{{1+\|x\|_2^2}^{\frac{3}{2}}}xx^T + \frac{1}{(1+\|x\|_2^2)^{\frac{1}{2}}}I$$ I see that $$\|A(x)\|_2 \le 1 \ \forall x$$

But is $\|A(x)\| \le 1$ in general $\forall x$ and if so, how to show this?.