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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. 

in Linear Algebra by Expert (2.4k points) | 94 views

1 Answer

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Best answer

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.

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