AA* = |A|E.
|A*| = |A|^(n-1)
当 r(A) = n 时, r(A*) = n
当 r(A) = n-1 时, r(A*) = 1
当 r(A) < n-1 时, r(A*) = 0
证明:
A*(A*)* = |A*|E
AA*(A*)* = |A*|A
|A| (A*)* = |A|^(n-1) A
所以, 当A可逆时, (A*)* = |A|^(n-2) A.
当A不可逆时, |A|=0
r(A) <= n-1.
r(A*)<= 1.
r((A*)*) = 0
即有 (A*)* = 0 = |A|^(n-2) A
