Functions of Matrices

Definition 1. Let $A \in M_{n}(F)$. Then we define
$$e^A = \sum_{n=0}^{\infty} \frac{1}{n!} A^n, \\ \sin(A) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} A^{2n+1}, \\ \cos(A) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} A^{2n}.$$