Given that the Eigen values of matrix A_3×3 are 1, 3 and 5. The Eigen values of A³ are:

Explanation:

Eigenvalues Analysis

Definition: Eigenvalues are scalar values associated with a square matrix that satisfy the equation A × v = λ × v, where A is the matrix, λ is the eigenvalue, and v is the eigenvector. Eigenvalues represent the scaling factors of the eigenvectors during linear transformations.

Key Concept: If λ is an eigenvalue of a matrix A, then λⁿ is an eigenvalue of the matrix Aⁿ, where n is a positive integer. This property arises from the fact that eigenvalues maintain their relationship with the matrix's powers during successive matrix multiplications.

Solution:

Given the eigenvalues of matrix A as 1, 3, and 5:

• For A³, the eigenvalues are calculated as λ³, where λ are the eigenvalues of A.
• Thus, the eigenvalues of A³ are:
  • (1)³ = 1
  • (3)³ = 27
  • (5)³ = 125

Therefore, the eigenvalues of A³ are 1, 27, and 125.