(v)显然有R(A4)cR(A2N(A)N(4) X rank(A)=rank(AA)=rank(A A) R(AA=R(A), N(AA)=N(A) 推论2设A∈Cm,则 (i)rmk(A)=n的充要条件是A=En; (i) rank(4)=m的充要条件是4=Em 证:充分性:定理3(1→rmk(A)=rmk(A) rank(en=nR(AA ) = R(A), N(A A) = N(A) − − (v) R(AA )  R(A), N(A A)  N(A) 显然有 − − rank(A) rank(AA ) rank(A A) − − 又 = = ( ) ( ) ; A n A A En i rank = = 的充要条件是 − 推论2 设 AC mn ,则 (i i) rank(A) = m AA = Em. 的充要条件是 − 充分性：定理3 (ii) ( ) ( ) − rank A = rank AA = rank(En ) = n 证: