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.