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If 2 and 4 are the eigen values of A then the eigenvalues of AT are
- a)1/2, 1/4
- b)2, 4
- c)4, 16
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If 2 and 4 are the eigen values of A then the eigenvalues of AT area)1...
The eigenvalues of a matrix A are the roots of the characteristic equation of A, which are invariant under the transpose operation. This means that the matrix A and its transpose AT have the same eigenvalues.
Given that 2 and 4 are the eigenvalues of A, the eigenvalues of AT will also be 2 and 4.
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If 2 and 4 are the eigen values of A then the eigenvalues of AT area)1...
Solution:
Given, 2 and 4 are the eigenvalues of matrix A
We need to find the eigenvalues of matrix AT
Let v be the eigenvector of A corresponding to eigenvalue 2
then we have
Av = λv
Multiplying both sides by AT, we get
(AT)Av = (AT)λv
⇒ (AT)A v = λ(AT)v
Hence, (AT)v is an eigenvector of AT with eigenvalue λ = 2.
Similarly, let u be the eigenvector of A corresponding to eigenvalue 4
then we have
Au = λu
Multiplying both sides by AT, we get
(AT)Au = (AT)λu
⇒ (AT)A u = λ(AT)u
Hence, (AT)u is an eigenvector of AT with eigenvalue λ = 4.
Therefore, the eigenvalues of matrix AT are 2 and 4.
Hence, the correct answer is option B.
Given, 2 and 4 are the eigenvalues of matrix A
We need to find the eigenvalues of matrix AT
Let v be the eigenvector of A corresponding to eigenvalue 2
then we have
Av = λv
Multiplying both sides by AT, we get
(AT)Av = (AT)λv
⇒ (AT)A v = λ(AT)v
Hence, (AT)v is an eigenvector of AT with eigenvalue λ = 2.
Similarly, let u be the eigenvector of A corresponding to eigenvalue 4
then we have
Au = λu
Multiplying both sides by AT, we get
(AT)Au = (AT)λu
⇒ (AT)A u = λ(AT)u
Hence, (AT)u is an eigenvector of AT with eigenvalue λ = 4.
Therefore, the eigenvalues of matrix AT are 2 and 4.
Hence, the correct answer is option B.
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