Let and be 2 non zero column vectors. Let denote the transpose of . Let . What is the rank of A?
What do you mean by , because we don't know what "y" is! I think maybe you're referring to the case where x=y. And I think in this case will have the full column rank if , and it will have the full row rank if .
Since is square, and it is an nxn matrix, it follows that is invertible if and only if has rank n. However has the same rank as x itself, so is invertible if rank(x)=n (this means if x has full column rank).
Again I'm assuming x=y.
I see no reason to assume that x= y. The problem makes sense without that. For example, if and , then and
. The determinant of that is is acbd- adbc= 0 so it does NOT have rank 2. It is not, unless at least one of x and y is the 0 vector, all "0"s so it does not have rank 0. It has rank 1.