X = [x1, x2, … xn]T is an n-tuple nonzero vector. The n × n matrix V = XXT A. has rank zero B. has rank 1 C. is orthogonal D. has rank n

The correct answer is: D. has rank n.

The rank of a matrix is the number of linearly independent rows or columns in the matrix. A matrix has rank n if it has n linearly independent rows or columns.

In this case, X is an n-tuple nonzero vector. This means that X has n linearly independent rows. Therefore, the n × n matrix V = XXT has rank n.

Option A is incorrect because a matrix with rank zero has no linearly independent rows or columns. Option B is incorrect because a matrix with rank 1 has only one linearly independent row or column. Option C is incorrect because an orthogonal matrix is a square matrix with orthonormal columns. In this case, X is an n-tuple nonzero vector, so XXT is not necessarily an orthogonal matrix.