Proof: 3λ is an Eigenvalue of 3A

In summary, if λ is an eigenvalue of the matrix A, then 3λ is also an eigenvalue of 3A. This can be proven by showing that 3(Ax)=λ(3x), where x is a non-zero vector, and thus, 3x cannot equal to 0. Therefore, 3λ is an eigenvalue of 3A.
  • #1
sana2476
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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?
 
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  • #2
Hi sana2476! :wink:
sana2476 said:
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. :smile:​
 
  • #3
Then what do think I should work with to prove that 3λ is an e.v of 3A?
 
  • #4
What is the formula for "3λ is an e.v of 3A?" :wink:
 

Related to Proof: 3λ is an Eigenvalue of 3A

1. What does it mean for 3λ to be an Eigenvalue of 3A?

When we say that 3λ is an Eigenvalue of 3A, it means that 3λ is a possible scalar value that can be multiplied by a vector to get the same vector back again after multiplying it by the matrix 3A. In other words, 3A transforms the vector in such a way that it is just a scaled version of itself, with the scaling factor being 3λ.

2. How is the Eigenvalue 3λ related to the matrix 3A?

The matrix 3A is said to have 3λ as an Eigenvalue because when we multiply 3A by a vector and get back the same vector scaled by 3λ, it satisfies the definition of an Eigenvalue. This means that 3A has a special property where it transforms certain vectors into scaled versions of themselves, with the scaling factor being the Eigenvalue 3λ.

3. Can there be more than one Eigenvalue for a matrix?

Yes, a matrix can have multiple Eigenvalues. The number of Eigenvalues a matrix has is equal to the number of dimensions in the vector space it operates on. In this case, the matrix 3A operates on a 3-dimensional vector space, so it can have up to 3 Eigenvalues.

4. How is the Eigenvalue 3λ calculated?

The Eigenvalue 3λ is calculated by solving the characteristic equation det(3A - 3λI) = 0, where det() is the determinant function, 3A is the matrix, 3λ is the scalar value, and I is the identity matrix. This equation will give us the possible values of 3λ that satisfy the definition of an Eigenvalue.

5. What is the significance of 3λ being an Eigenvalue of 3A?

The significance of 3λ being an Eigenvalue of 3A is that it tells us about the special properties of the matrix 3A. It means that 3A has a special transformation property where it can scale certain vectors by a factor of 3λ. This information can be useful in understanding the behavior of the matrix and its effect on vectors in the vector space it operates on.

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