## Acalytica

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For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to justify your answers to this problem - but you should know either a proof or a counterexample.

a) If $A$ and $B$ are $4 \times 4$ matrices such that $\operatorname{rank}(A B)=3$, then $\operatorname{rank}(B A)<4$.
b) If $A$ is a $5 \times 3$ matrix with $\operatorname{rank}(A)=2$, then for every vector $b \in \mathbb{R}^{5}$ the equation $A x=b$ will have at least one solution.
c) If $A$ is a $4 \times 7$ matrix, then $A$ and $A^{T}$ have the same rank.
d) Let $A$ and $B \neq 0$ be $2 \times 2$ matrices. If $A B=0$, then $A$ must be the zero matrix.
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