In mathematics, determinant is a function of matrix A whose domain is det, and its value is scalar, which is denoted as det(A) or |A|. Whether in linear algebra, polynomial theory or calculus (such as substitution integral method), determinant, as a basic mathematical tool, has important applications.

When the determinant is equal to zero, it can be concluded that |A|=0:

1 is linearly related to the line vector of a;

2. The column vectors of A are linearly related;

3. The equation group Ax=0 has a nonzero solution;

4. the rank of a is less than n (n is the order of a)

5.a is irreversible

The determinant of a matrix is equal to and not equal to 0 |A|≠0.

& lt=> Reversible (non-singular)

& lt=> There exists a square matrix B of the same order that satisfies AB=E (or BA=E).

& lt= & gtR(A)=n

The column (row) vector group of<=>A is linearly independent.

& lt= & gtAX=0 has only zero solution.

& lt= & gtAX=b has a unique solution.

& lt=> any n-dimensional vector can be uniquely and linearly represented by a set of column vectors of a.

& lt=>a can be expressed as the product of elementary matrices.

The equivalent canonical form of<=>A is identity matrix.

The simplest form of the<=>A line is identity matrix.

The eigenvalue of<=>a is not equal to 0.

& lt= & gtA^TA is a positive definite matrix.