Calculation
If A is a square matrix, the determinant of A, det(A), is defined to be the sum of all signed elementary products from A.
Basic properties of the determinant function
Suppose that A and B are n × n matrices and k is scalar.
If A and B are square matrices of the same size, then
If A is invertible, then
Determinant of a transposed matrix
If A is a square matrix, then
Determinant of a triangular matrix
If A is an n × n triangular matrix,
Determinant from LU decomposition
For matrices larger than 3×3, it is computationally cheaper (in number of operations conducted) to express, if possible, A = LU, where L is a lower triangular matrix with diagonal elements equal to 1, and U is an upper triangular matrix.
because det(L) = 1; the right hand side is easily computed as the product of all diagonal elements of U.
A small example:
