<s>[INST] Let $A$ be a square matrix with dimensions $n \times n$. Let $B$ be a square matrix with dimensions $m \times m$. Let $C$ be the square matrix with dimensions $n \times n$ such that $C = AB$, where $A$ and $B$ satisfy the following conditions:
1) Matrix $A$ is singular.
2) Matrix $B$ is singular.
3) Matrix $A$ is symmetric and idempotent.
4) Matrix $B$ is diagonal.

From the given conditions, we can conclude that:

I. $C$ is singular.
II. $C$ is idempotent.
III. $C$ is diagonal.

Which of the following is correct?

A) I and II
B) I and III
C) I only
D) II only
E) III only[/INST]