Let \(A\) and \(B\) be \(n \times n\) matrices.
a) Show that the \(\operatorname{rank}(A B) \leq \operatorname{rank}(A)\). Give an example where strict inequality can occur.
b) Show that \(\operatorname{dim}(\operatorname{ker} A B) \geq \operatorname{dim}(\operatorname{ker} A)\). Give an example where strict inequality can occur.