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If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen values of A2 are
- a)-1 , 2 , 3
- b)1, 4, 9
- c)1, 2, 3
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen...
If λ1 ,λ2 ,λ3 ....λn are the eigen values of a matrix A, then A2 has the eigen values λ12 ,λ22 ,λ32 ....λn2 So, eigen values of A2 are 1, 4, 9.
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If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen...
Solution:
Given, the eigen values of a square matrix A are -1, 2 and 3.
We need to find the eigen values of A^2.
Let λ be an eigen value of A and x be the corresponding eigen vector.
Then, Ax = λx
Multiplying both sides by A, we get
A^2x = A(λx) = λAx = λ(λx) = λ^2x
So, λ^2 is an eigen value of A^2 corresponding to the eigen vector x.
Therefore, the eigen values of A^2 are (λ1)^2, (λ2)^2, (λ3)^2, where λ1, λ2 and λ3 are the eigen values of A.
Substituting the given eigen values, we get
(λ1)^2 = (-1)^2 = 1
(λ2)^2 = 2^2 = 4
(λ3)^2 = 3^2 = 9
Hence, the eigen values of A^2 are 1, 4 and 9, which is option (B).
Given, the eigen values of a square matrix A are -1, 2 and 3.
We need to find the eigen values of A^2.
Let λ be an eigen value of A and x be the corresponding eigen vector.
Then, Ax = λx
Multiplying both sides by A, we get
A^2x = A(λx) = λAx = λ(λx) = λ^2x
So, λ^2 is an eigen value of A^2 corresponding to the eigen vector x.
Therefore, the eigen values of A^2 are (λ1)^2, (λ2)^2, (λ3)^2, where λ1, λ2 and λ3 are the eigen values of A.
Substituting the given eigen values, we get
(λ1)^2 = (-1)^2 = 1
(λ2)^2 = 2^2 = 4
(λ3)^2 = 3^2 = 9
Hence, the eigen values of A^2 are 1, 4 and 9, which is option (B).
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If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen...
Eigenvalues of a Matrix A:
The eigenvalues of a square matrix A are the values λ for which there exists a non-zero vector v such that Av = λv. In other words, the eigenvalues of A are the values that satisfy the equation A - λI = 0, where I is the identity matrix.
Eigenvalues of A^2:
To find the eigenvalues of A^2, we need to calculate the eigenvalues of the matrix A^2. Since A^2 = A * A, we can rewrite the equation as (A * A - λI) = 0.
Understanding the Eigenvalues of A^2:
The eigenvalues of A^2 are the values λ for which there exists a non-zero vector v such that (A * A - λI) * v = 0. This can be further simplified to (A * (A * v) - λ * v) = 0.
Solving for Eigenvalues of A^2:
To find the eigenvalues of A^2, we substitute the given eigenvalues of A into the equation (A * (A * v) - λ * v) = 0.
For each eigenvalue λ, we solve the equation (A * (A * v) - λ * v) = 0 to find the corresponding eigenvectors v. Then, we calculate A^2 * v for each eigenvector v to find the eigenvalues of A^2.
Solution:
Given eigenvalues of A: -1, 2, 3
1. For λ = -1:
We solve the equation (A * (A * v) - (-1) * v) = 0 to find the eigenvectors v. Let's assume v1 is an eigenvector corresponding to λ = -1. We calculate A^2 * v1 to find the eigenvalue of A^2 corresponding to λ = -1.
2. For λ = 2:
We solve the equation (A * (A * v) - 2 * v) = 0 to find the eigenvectors v. Let's assume v2 is an eigenvector corresponding to λ = 2. We calculate A^2 * v2 to find the eigenvalue of A^2 corresponding to λ = 2.
3. For λ = 3:
We solve the equation (A * (A * v) - 3 * v) = 0 to find the eigenvectors v. Let's assume v3 is an eigenvector corresponding to λ = 3. We calculate A^2 * v3 to find the eigenvalue of A^2 corresponding to λ = 3.
After calculating A^2 * v1, A^2 * v2, and A^2 * v3, we obtain the following eigenvalues of A^2: 1, 4, 9.
Therefore, the correct option is b) 1, 4, 9.
The eigenvalues of a square matrix A are the values λ for which there exists a non-zero vector v such that Av = λv. In other words, the eigenvalues of A are the values that satisfy the equation A - λI = 0, where I is the identity matrix.
Eigenvalues of A^2:
To find the eigenvalues of A^2, we need to calculate the eigenvalues of the matrix A^2. Since A^2 = A * A, we can rewrite the equation as (A * A - λI) = 0.
Understanding the Eigenvalues of A^2:
The eigenvalues of A^2 are the values λ for which there exists a non-zero vector v such that (A * A - λI) * v = 0. This can be further simplified to (A * (A * v) - λ * v) = 0.
Solving for Eigenvalues of A^2:
To find the eigenvalues of A^2, we substitute the given eigenvalues of A into the equation (A * (A * v) - λ * v) = 0.
For each eigenvalue λ, we solve the equation (A * (A * v) - λ * v) = 0 to find the corresponding eigenvectors v. Then, we calculate A^2 * v for each eigenvector v to find the eigenvalues of A^2.
Solution:
Given eigenvalues of A: -1, 2, 3
1. For λ = -1:
We solve the equation (A * (A * v) - (-1) * v) = 0 to find the eigenvectors v. Let's assume v1 is an eigenvector corresponding to λ = -1. We calculate A^2 * v1 to find the eigenvalue of A^2 corresponding to λ = -1.
2. For λ = 2:
We solve the equation (A * (A * v) - 2 * v) = 0 to find the eigenvectors v. Let's assume v2 is an eigenvector corresponding to λ = 2. We calculate A^2 * v2 to find the eigenvalue of A^2 corresponding to λ = 2.
3. For λ = 3:
We solve the equation (A * (A * v) - 3 * v) = 0 to find the eigenvectors v. Let's assume v3 is an eigenvector corresponding to λ = 3. We calculate A^2 * v3 to find the eigenvalue of A^2 corresponding to λ = 3.
After calculating A^2 * v1, A^2 * v2, and A^2 * v3, we obtain the following eigenvalues of A^2: 1, 4, 9.
Therefore, the correct option is b) 1, 4, 9.
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If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen values of A2 area)-1 , 2 , 3b)1, 4, 9c)1, 2, 3d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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