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Consider an n×n matrix A and a non-zero n×1 vector p. Their product Ap = α2 p, where {α ∈ ℜ ∉and α -1,0,1} . Based on the given information, the Eigen value of A2 is :
- a)α
- b)α2
- c)√α
- d)α4
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Consider an n×n matrix A and a non-zero n×1 vector p. Thei...
Given : AP = α2P
On comparison with AX =λX i.e.
Eigen value of A is α2
So, Eigen value of A2 is (α 2)2= α4
Hence, the correct option is (D).
On comparison with AX =λX i.e.
Eigen value of A is α2
So, Eigen value of A2 is (α 2)2= α4
Hence, the correct option is (D).
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Consider an n×n matrix A and a non-zero n×1 vector p. Thei...
Explanation:
Given Information:
- Matrix A is a n×n matrix.
- Vector p is a non-zero n×1 vector.
- Ap = α^2 p, where α ∈ ℝ and α ≠ -1, 0, 1.
Finding Eigenvalues of A^2:
- Let λ be an eigenvalue of A with corresponding eigenvector v.
- From the given information, Ap = α^2 p.
- This implies that Av = α^2 v (since Ap = Av when p = v).
- Therefore, α^2 is an eigenvalue of A^2, as the eigenvalues of A^2 are the squares of the eigenvalues of A.
Conclusion:
- The eigenvalues of A^2 are the squares of the eigenvalues of A.
- As given, α is an eigenvalue of A. Therefore, α^2 is an eigenvalue of A^2.
- Hence, the correct answer is option 'D' which is α^4.
Given Information:
- Matrix A is a n×n matrix.
- Vector p is a non-zero n×1 vector.
- Ap = α^2 p, where α ∈ ℝ and α ≠ -1, 0, 1.
Finding Eigenvalues of A^2:
- Let λ be an eigenvalue of A with corresponding eigenvector v.
- From the given information, Ap = α^2 p.
- This implies that Av = α^2 v (since Ap = Av when p = v).
- Therefore, α^2 is an eigenvalue of A^2, as the eigenvalues of A^2 are the squares of the eigenvalues of A.
Conclusion:
- The eigenvalues of A^2 are the squares of the eigenvalues of A.
- As given, α is an eigenvalue of A. Therefore, α^2 is an eigenvalue of A^2.
- Hence, the correct answer is option 'D' which is α^4.
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Consider an n×n matrix A and a non-zero n×1 vector p. Their product Ap = α2 p, where {α ∈ ℜ ∉and α -1,0,1} . Based on the given information, the Eigen value of A2 is :a)αb)α2c)√αd)α4Correct answer is option 'D'. Can you explain this answer?
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