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## Homework Statement

Find the eigenvalues/vectors of A. (I can do this bit :P, A is a 3x3 matrix)

What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is

the identity matrix?

## Homework Equations

## The Attempt at a Solution

I have (correctly) found that A has eigenvectors and corresponding eigenvalues such that

A|1>=|1>, A|2>=2|2>, A|4>=4|4>.

|1>=(1,1,1), |2>=(1,-1,0), |4>=(1,1,-2) though i don't actually think you need this.

as B can be expanded in a power series of A, and A commutes with the identity (obiously)

[B,A]=0

=> |1>, |2>, |4> are eigenvectors of B

**I am not confident in the reasoning that follows, does it seem correct?**

exp(3A)|j>=e^(3j)|j>

5I|j>=5|j>

j=1,2,4

=> the eigenvectors of B are |1>, |2>, |4> with respective eigenvalues (e^3 +5), (e^6 +5), (e^12 +5)

**Does this seem correct? Thanks in advance for your help.**