Proof: 3λ is an Eigenvalue of 3A

[sana2476]

Homework Statement

If λ is and eigenvalue of the the matrix A then 3λ is an eigenvalue of 3A

Homework Equations

The Attempt at a Solution

. .
. λ is an e.v of A

Therefore, ∃ x not equal to 0 s.t Ax=λx
Then, 3Ax=3λx
which can written as 3(Ax)=3(λx)=λ(3x)
and 3x does not equal to 0 because x doesn't equal to zero and obviously neither does 3.

Therefore, we can conclude that 3λ is an eigenvalue of 3A.

This was my attempt at the proof. However, I'm not sure if it suffices to conclude that neither 3 nor x equal to zero. Is there anything else I need to add to complete this proof?

 

[tiny-tim]

Hi sana2476!

3(Ax)=3(λx)=λ(3x)
and 3x does not equal to 0 because x doesn't equal to zero and obviously neither does 3.

Therefore, we can conclude that 3λ is an eigenvalue of 3A.

No. 3(Ax) = λ(3x) doesn't prove anything.

Try again. ​

 

[sana2476]

Then what do think I should work with to prove that 3λ is an e.v of 3A?

 

[tiny-tim]

What is the formula for "3λ is an e.v of 3A?"