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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.
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