|A*|=0

|A|=0

Obviously established;

Irreversible

A*=|A|A^(- 1)

Take the determinant and get

|a*|=||a|a^(- 1)|=|a|^zhin |a^(- 1)|

=|A|^n |A|^(- 1)

=|A|^(n- 1)

Correlation theorem

Theorem 1. Let a be an n×n matrix, then det(AT)=det(A)[2].

The proof of n is proved by mathematical induction. Obviously, this conclusion holds for n= 1 because the matrix of1is symmetric. Suppose this conclusion is also true for all k×k matrices. For (k+ 1)×(k+ 1) matrix A, expand det(A) according to the first line of A, we have:

det(A)= A 1 1 det(m 1 1)-A 12 det(m 12)+-…A 1，k+ 1det(M 1，k+ 1).

Theorem 2. Let a be an n×n triangular matrix. Then the determinant of a is equal to the product of diagonal elements of a.

According to the theorem 1, it is only necessary to prove that the conclusion holds for the lower triangular matrix. This conclusion can be easily proved by using the expansion of cofactor and the induction of n.