subspace and dimension

Question

Let $V$ be a subspace of $\mathbb{R}^{10}$. Suppose $A$ is a $10 \times 10$ matrix with real entries. Let $A^k(V) = \{A^k\mathbf{x} : \mathbf{x} \in V\}$ for $k \ge 1$ and $A(V) = A^1(V)$. Which one of the following statements is NOT true?

MCQ2

(A) If $A$ is nonsingular, then $\dim(V) = \dim(A(V))$ necessarily holds

(B) It is possible that $A$ is singular and $\dim(V) = \dim(A(V))$

(C) If $\operatorname{rank}(A) = 8$, then $\dim(A(V)) \ge \dim(V) - 2$ necessarily holds

(D) If $\dim(V) = \dim(A(V)) = \dim(A^2(V)) = \cdots = \dim(A^5(V))$, then $\dim(A^6(V)) = \dim(V)$ necessarily holds