Explanation:
To determine whether matrix M is diagonalizable or nilpotent, we need to understand the definitions and properties of these terms.

Diagonalizable Matrix:
A square matrix M of order n is diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that M = PDP^(-1), where D is a diagonal matrix.

Nilpotent Matrix:
A square matrix M of order n is nilpotent if there exists a positive integer k such that M^k is the zero matrix (all entries are zero).

Rank of a Matrix:
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It can also be defined as the dimension of the row or column space of the matrix.

Given Information:
We are given that the rank of matrix M is 1.

Analysis:
Let's analyze the possible options one by one:

Option A: Diagonalizable and Non-Singular
If M is diagonalizable, then M can be written as M = PDP^(-1), where D is a diagonal matrix. Since the rank of M is 1, it implies that M has only one linearly independent row or column. In this case, the diagonal matrix D would have only one non-zero entry on the diagonal. However, since M is non-singular (invertible), it should have a non-zero determinant. But if D has only one non-zero entry, the determinant of D would be zero, implying that M is singular. Hence, option A is incorrect.

Option B: Diagonalizable and Nilpotent
If M is diagonalizable, then M can be written as M = PDP^(-1), where D is a diagonal matrix. Since the rank of M is 1, it implies that M has only one linearly independent row or column. In this case, the diagonal matrix D would have only one non-zero entry on the diagonal. However, if M is nilpotent, there exists a positive integer k such that M^k is the zero matrix. Since D is a diagonal matrix with only one non-zero entry, raising D to any power k would still result in a diagonal matrix with the same non-zero entry. Thus, M cannot be nilpotent. Hence, option B is incorrect.

Option C: Neither Diagonalizable nor Nilpotent
If M is neither diagonalizable nor nilpotent, it means that M does not satisfy the conditions for either of these properties. The given information about the rank of M being 1 does not provide any direct information about whether M is diagonalizable or nilpotent. Hence, option C is a possibility.

Option D: Either Diagonalizable or Nilpotent but not both
Since options A and B are incorrect, and option C is a possibility, the correct answer is option D. The given information does not provide enough information to determine whether M is diagonalizable or nilpotent.

Conclusion:
Based on the given information, the matrix M can either be diagonalizable or nilpotent, but it cannot be both. The exact nature of M cannot be determined solely from the given information.