If \(A\) and \(B\) are \(4 \times 4\) matrices such that \(\operatorname{rank}(A B)=3\), then \(\operatorname{rank}(B A)<4\).

Question

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.