section_path: Endomorphisms and square matrices/Determinant
section_name: Determinant
paragraph_text: Determinant
  
  The determinant of a square matrix A is defined to be
  
  where S is the group of all permutations of n elements, σ is a permutation, and (−1) the parity of the permutation. A matrix is invertible if and only if the determinant is invertible (i.e., nonzero if the scalars belong to a field).
  
  Cramer's rule is a closed-form expression, in terms of determinants, of the solution of a system of n linear equations in n unknowns. Cramer's rule is useful for reasoning about the solution, but, except for n = 2 or 3, it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm.
  
  The determinant of an endomorphism is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense, since this determinant is independent of the choice of the basis.