五、几个常用的公式 1) AA*=AA=AE 2) A*=AA1 3) 14-1=11 4) 124=A 5) (2A)1=1A1 例4若4≠0，试证(1)41=4-1，（2)(4)1=(4) (3)(A)T=(AT);(4)(A*)*=A-2A;(5)(k4)=-1A。 证(1)A1=AM=Am4=A-: (2)(A*1=(AA1>1=A(A)1=(A); (3)(A)T=(AA)T=AT(41)T=A(A)1=(A) 五、几个常用的公式 1) AA* = A*A = |A|E 2) A* = |A|A-1 3) |A-1 | = |A| -1 4) |λA| = λ n |A| 5) (λA) -1 = λ -1A-1 例4 若 |A| ≠ 0, 试证（1） |A* | =|A| n-1 ;（2）(A* ) -1= (A-1 ) * (3) (A* ) T = (AT ) * ;（4）(A*)* = |A| n-2A;（5）(kA) * = k n-1A* 。 证 （1） |A* | = (2) (A* ) -1= (3) (A* ) T = ||A|A-1 | = |A| n |A-1 | = |A| n-1 ; (|A|A-1 ) -1 = |A-1 |(A-1 ) -1 = (A-1 ) * ; ( |A|A-1 ) T = |AT|(A-1 ) T = |AT|(AT) -1 = (AT ) *