Let $A$ and $B$ be $n \times n$ matrices with real entries.
Consider the following statements:
P: If $A$ is symmetric then $\operatorname{rank}(A)=$ Number of nonzero eigenvalues (counting multiplicity) of $A$.
Q: If $A B=\mathbf{0}$ then $\operatorname{rank}(A)+\operatorname{rank}(B) \leq n$
Then
- both $\mathrm{P}$ and $\mathrm{Q}$ are TRUE
- $\mathrm{P}$ is TRUE and $\mathrm{Q}$ is FALSE
- $\mathrm{P}$ is FALSE and $\mathrm{Q}$ is TRUE
- both $\text{P}$ and $\text{Q}$ are FALSE
