Suppose X is n×p with p ≤ n. If X has rank p, show that X

X has rank p, and conversely. Hints. Suppose X has rank p and c is p×1. Then X

Xc = 0p×1 ⇒ c

X

Xc = 0 ⇒ Xc2 = 0 ⇒ Xc = 0n×1.

Notes. The matrix X

X is p×p. The rank is p if and only if X

X is invertible. The ⇒ is shorthand for “implies.”