Re: [理工] [線代]- rank跟adjoint的問題

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※ 引述《mailmovieb (  )》之銘言：
: 我想問說如果要證明
: A是n階方陣，n>1
: 1.rank(A)＝n   if and only if rank(adj(A))=n
: 2.rank(A)＝n-1 if and only if rank(adj(A))=1
: 3.rank(A)＜n-1 if and only if adj(A)=0
: 請問該如何證明？
pf:

1. A:nonsingular <=> det(A) != 0
<=> det(adj(A)) = det(A)^(n-1) !=0
<=> adj(A): nonsingular <=> rank(adj(A)) = n

3. Suppose Aij is the cofactor of A, for any i,j,
since Aij is the determinant of a n-1 * n-1 submatrix S, then
rank(A) ＜ n-1 <=> S is singular, for all S
<=> det(S) = 0, for all S
<=> all entries of adj(A) are equal to zero
<=> adj(A) = 0 <=> rank(adj(A)) = 0

2.
(=>)
Since det(A) = 0 => A(adj(A)) = 0
=> R(adj(A)) is contained in N(A)
=> rank(adj(A)) ≦ nullity(A) = n - rank(A) = 1
since rank(A)=n-1, applying 3 we can get rank(adj(A)) != 0,
therefore rank(adj(A)) = 1

(<=)
we can know that rank(A) ≧ n-1 but is not equal to n (by 1 and 3),
rank(A) can only be n-1.

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