Linear Algebra Eigenvector Properties

[FinalStand]

Homework Statement

True/False: If true give a proof, if false give a counterexample.
a)
If A and B have the same eigenvector X, then A+B should also have the same eigenvector, X.
b)
if A has an eigenvalue of 2, and B has an eigenvalue of 5, then 7 is an eigenvalue of A+B

Homework Equations

The Attempt at a Solution

for b):
2 0 2 3
0 2 has eigenvalue of 2; 3 2 has eigenvalue of 5

When I add them together (A+B) you get 4 3
3 4

Then I found an eigenvalue of 7; Is this correct?
Or the property of A+B != eigenvalueA + eigenvalueB is always correct? But this question's wording is kind of weird, because it said if its true give a counterexample ...


for a) I think it is false,...not entirely sure though.

 

[epenguin]

True/False: If true give a proof, if false give a counterexample.



But this question's wording is kind of weird, because it said if its true give a counterexample ...

That's not what I read.

 

[FinalStand]

Ok I am stupid, I read the question wrong so I confused myself...here goes my mark...

 

[HallsofIvy]

(a) asks you to show that If X is an eigenvector for both A and B then it is an eigenvector for A+ B. If X is an eigevector of A, then $AX= \lambda_A X$ for some number $\lambda_A$. If X is an eigenvector of B, then $BX= \lambda_B X$ for some number $\lambda_B$. Now, what can you say about (A+ B)X?