Let denote the set of all matrices with entries in such that one row consists of all 0s. Show that is a semigroup under matrix multiplication and exhibit its idempotents.
The elements of come in two flavors, and , where and . Clearly . (If is vanilla and is chocolate, then the zero matrix is swirl.)
Since is a subset of the known semigroup under multiplication, it suffices to show that is closed. To that end, it suffices to consider four cases. In what follows, let . (Note that is closed under multiplication.)
So is indeed a semigroup.
Suppose now that is a nonzero idempotent. Then we have , and comparing entries, and . Now in , so that either or . But if , then , so . Thus . Indeed, is idempotent.
Similarly, if is a nonzero idempotent, then , and conversely.
So the idempotents in are either 0, for some , or for some .