Question

Let the eigenvalues of a 2 × 2 matrix A be 1, -2 with eigenvectors x₁ and x₂ respectively. Then the eigenvalues and eigenvectors of the matrix A³ would, respectively, be
1. 1, -8: x₁, x₂
2. -1, -2: x₁ + x₂, x₁ - x₂;
3. 1, -2: x₁, x₂
4. 2, 0: x₁ + x₂, x₁ - x₂;

Answer 1

Correct Answer - Option 1 : 1, -8: x₁, x₂

Concept:

If λ is an eigenvalue of a matrix A and k  is a scalar then:

1) λm is the eigenvalue of matrix Am (m belongs to N).

2) kλ  is an eigenvalue of matrix kA.

3) λ + k is an eigenvalue of the matrix A + kI.

4) λ - k  is an eigenvalue of matrix A - kI.

Calculation:

For a given matrix, if the eigenvalues are λ₁ and λ₂, then the eigenvalues of Aⁿ will be \(λ_1^n \; and \; λ_2 ^n \).

1 and -2 will become 1 and -8;

Although the eigenvectors remain the same.