Statement: If A is a 3 x 3 matrix that satisfies A^3 = A, then A is diagonalizable.

Explanation:

Diagonalization of a matrix refers to the process of finding a diagonal matrix D and an invertible matrix P such that A = PDP^(-1), where D is a diagonal matrix with eigenvalues of A on its diagonal.

To determine if A is diagonalizable, we need to consider its eigenvalues and eigenvectors.

Eigenvalues:

The characteristic polynomial of a matrix A is defined as det(A - λI), where λ is a scalar and I is the identity matrix. The eigenvalues of A are the values of λ that satisfy this characteristic polynomial.

In this case, we have A^3 = A. Let's consider the characteristic polynomial of A:

det(A - λI) = det(A^3 - λI) = det(A^3 - A) = det(A(A^2 - I))

Since A^3 = A, we can simplify the above expression as:

det(A(A^2 - I)) = det(A(A - I)(A + I))

The eigenvalues of A are the values of λ that satisfy det(A(A - I)(A + I)) = 0.

Eigenvectors:

For each eigenvalue λ, the eigenvectors of A are the non-zero vectors x that satisfy the equation Ax = λx.

To determine if A is diagonalizable, we need to check if there are enough linearly independent eigenvectors for each eigenvalue.

Explanation for option A:

If A is a 3 x 3 matrix that satisfies A^3 = A, then we can conclude the following:

- The eigenvalues of A are the values of λ that satisfy det(A(A - I)(A + I)) = 0.
- The eigenvectors of A can be found by solving the equation Ax = λx for each eigenvalue.

Since A satisfies A^3 = A, it follows that the characteristic polynomial of A is different from its minimal polynomial. This implies that A has distinct eigenvalues.

If A has distinct eigenvalues, then it can be diagonalized. This means that there exists an invertible matrix P and a diagonal matrix D such that A = PDP^(-1). Therefore, option A is correct.