设A为n（n＞2）阶方阵,证明A可逆的充分必要条件是A*可逆

Question

设A为n（n＞2）阶方阵,证明A可逆的充分必要条件是A*可逆

Answer 1

AA* = |A|·E.
若A可逆,有|A| ≠ 0,A* = |A|·A^(-1)也是可逆的.
若A不可逆,有|A| = 0,故AA* = 0.
r(A)+r(A*)-n ≤ r(AA*) = 0,即r(A*) ≤ n-r(A).
当A ≠ 0,r(A) > 0,得r(A*) < n,A*不可逆.
而当A = 0,由伴随矩阵的构造易得A* = 0,A*同样不可逆.
实际上可以证明:对n > 1,
r(A) = n时,r(A*) = n.
r(A) = n-1时,r(A*) = 1.
r(A) < n-1时,r(A*) = 0.