Answer whether the following statements are True or False.
If $A \in \mathrm{M}_{n}(\mathbb{R})$ is such that $A B=B A$ for all invertible matrices $B \in \mathrm{M}_{n}(\mathbb{R})$, then $A=\lambda \cdot \text{Id}$ for some $\lambda \in \mathbb{R}.$