Given a $n\times n$ matrix $B$ defined by $e^B=\sum_{j=0}^\infty \frac{B^j}{j!}$. What is the matrix $e^{p(B)}$?

Question

Given a $n\times n$ matrix $B$ defined by $e^B$ by $$e^B=\sum_{j=0}^\infty \frac{B^j}{j!}$$

Let $p$ be the characteristic polynomial of $B$ Then the matrix $e^{p(B)}$ is

1. $I_{n\times n}$
2. $e \times I_{n\times n}$
3. $e \times I_{n\times n}$
4. $0_{n\times n}$
5. $\pi \times I_{n\times n}$

Choose the correct option from above. 

Answer 1

By Caley Hamilton Theorem, we know that every matrix satisfies its own characteristic polynomial. Thus, $p(B)=0$. This gives, 

$$e^0=I_{n\times n}$$  

The correct option is Item 1.