Basic Properties of Determinants

Basic Properties of Determinants 

Some basic properties of determinants are given below:

1. 
   If In is the identity matrix of the order m ×m, then det(I) is equal to1
   
2. 
   If the matrix X^T is the transpose of matrix X, then det (X^T) = det (X)
   
3. 
   If matrix X^-1 is the inverse of matrix X, then det (X^-1) = 1/det (x) = det(X)^-1 
   
4. 
   If two square matrices x and y are of equal  size, then det (XY) = det (X) det (Y)
   
5. 
   If matrix X retains size a × a and C is a constant, then det (CX) = C^a det (X)
   
6. 
   If A, B, and C are three positive semidefinite matrices of equal size, then the following equation holds along with the corollary det (A+B) ≥ det(A) + det (B) for A,B, C ≥ 0 det (A+B+C) + det C ≥ det (A+B) + det (B+C)
   
7. 
   In a triangular matrix, the determinant is equal to the product of the diagonal elements.
   
8. 
   The determinant of a matrix is zero if each element of the matrix is equal to zero.
   
9. 
   Laplace’s Formula and the Adjugate Matrix.