Question

Let A be a diagonalizable matrix whose eigen values satisfy that A2 = A + 1. Then A satisfies A) A² = A +1 B) A² = A + 2 C) A² = A D) A² = A + 1

Answer 1

D) A^2 = A + 1 this equation represents diagonal matrices with the same eigenvalues on the diagonal.

Let λ be an eigenvalue of the matrix A, and let v be the corresponding eigenvector. Since A is diagonalizable, we can write A = PDP^(-1), where D is the diagonal matrix containing the eigenvalues on the diagonal, and P is the matrix whose columns are the eigenvectors.

We know that A^2 = A + 1, so we can substitute A with its diagonalizable form:

(PDP^(-1))^2 = PDP^(-1) + 1.

Expanding the square and applying the matrix multiplication rules, we get:

PD^2P^(-1) = PDP^(-1) + 1.

Since D is a diagonal matrix, D^2 will have the eigenvalues squared on the diagonal. Therefore, we have:

P(D^2)P^(-1) = PDP^(-1) + 1.

Multiplying both sides by P^(-1) on the right, and by P on the left, we obtain:

D^2 = D + 1.

This equation holds because P^(-1)P = I (the identity matrix), and D and D^2 are diagonal matrices with the same eigenvalues on the diagonal.

Therefore, the correct answer is D) A^2 = A + 1.

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