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不需要A一定是可逆.
知識(shí)點(diǎn):
AA* = |A|E.
|A*| = |A|^(n-1)
當(dāng) r(A) = n 時(shí), r(A*) = n
當(dāng) r(A) = n-1 時(shí), r(A*) = 1
當(dāng) r(A) < n-1="" 時(shí),="" r(a*)="">
證明:
A*(A*)* = |A*|E
AA*(A*)* = |A*|A
|A| (A*)* = |A|^(n-1) A
所以, 當(dāng)A可逆時(shí), (A*)* = |A|^(n-2) A.
當(dāng)A不可逆時(shí), |A|=0
r(A) <=>
r(A*)<=>
r((A*)*) = 0
即有 (A*)* = 0 = |A|^(n-2) A