Indicate whether the statement is always true or sometimes false. Justify your answer with a logical argument

Question:

Indicate whether the statement is always true or sometimes false. Justify your answer with a logical argument or a counterexample.
(a) The expressions tr(AAT) and tr(ATA) are always defined, regardless of the size of A.
(b) tr(AAT) = tr (ATA) for every matrix A
(c) If the first column of A has all zeros, then so does the first column of every product AB.
(d) If the first row of A has all zeros, then so does the first row of every product AB.