Let the adjoint matrix of square matrix A of order n be a *, and prove that: (1) if | a | = 0, then | a * | = 0; （2）|A*|=|A|^(n-1) The first question can be asked by the inverse method

Let the adjoint matrix of square matrix A of order n be a *, and prove that: (1) if | a | = 0, then | a * | = 0; （2）|A*|=|A|^(n-1) The first question can be asked by the inverse method

(1)
Certificate:
If R (a)

Let a be a matrix of order n, and prove that R (a ^ n) = R (a ^ (n + 1))
linear algebra

If you know Jordan standard, it's obvious
If we don't know, we prove that a ^ {n + 1} x = 0 and a ^ n x = 0 are the same
If a is nonsingular, then it is obviously true, otherwise it is used
n-1 >= rank(A) >= rank(A^2) >= ... >= rank(A^n) >= rank(A^{n+1}) >=0
There must be two adjacent terms equal in the middle, that is, a ^ k x = 0 and a ^ {K + 1} x = 0 have the same solution, so a ^ {n + 1} x = 0 and a ^ n x = 0 have the same solution