If λ1, λ2, λ3 are the eigenvalues of a matrix A then A³ has the eigenvalues?
Understand the Problem
The question is asking about the eigenvalues of the matrix A and how they change when the matrix is raised to the power of 3 (A³). It presents multiple choice options regarding the resulting eigenvalues of A³ based on the original eigenvalues of A.
Answer
The eigenvalues of $A^3$ are $\lambda_1^3, \lambda_2^3, \lambda_3^3$.
Answer for screen readers
The eigenvalues of $A^3$ are $\lambda_1^3, \lambda_2^3, \lambda_3^3$.
Steps to Solve
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Understanding Eigenvalues Eigenvalues of a matrix $A$ are the scalars $\lambda$ such that there exists a non-zero vector $\mathbf{v}$ (the eigenvector) satisfying the equation $A\mathbf{v} = \lambda \mathbf{v}$.
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Eigenvalues of A³ If $\lambda_1, \lambda_2, \lambda_3$ are the eigenvalues of $A$, then the eigenvalues of the matrix $A^n$ (where $n$ is a positive integer) can be derived from the eigenvalues of $A$. Specifically, the eigenvalues of $A^3$ will be $\lambda_1^3, \lambda_2^3, \lambda_3^3$.
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Conclusion of the Analysis Based on the properties of eigenvalues, for the matrix $A^3$, we conclude that the eigenvalues of $A^3$ are $λ_1^3, λ_2^3, λ_3^3$.
The eigenvalues of $A^3$ are $\lambda_1^3, \lambda_2^3, \lambda_3^3$.
More Information
This result follows from the properties of eigenvalues in linear algebra. When a matrix is raised to a power, the eigenvalues are also raised to that power.
Tips
- Confusing eigenvectors and eigenvalues: Always remember that eigenvalues are scalars that correspond to eigenvectors which remain in the direction of the eigenvector after the transformation.
- Misapplying polynomial properties: Ensure to apply the correct power to each eigenvalue when calculating eigenvalues for matrix powers.
