Determinants

Permutations Permutations of the integers 1 through n are either even or odd. A permutation is even if it can be achieved by an even number of interchanges (transpositions), and odd if it takes an odd number of them. There are n! permutations. We define the function sign(p) to be +1 if p is an even permutation, and -1 if p is odd.

Definition of a determinant If A is an n×n matrix, then we define det(A) via

det(A) = Σpsign(p) a1,p1a2,p2 ...an,pn ,   p = (p1,p2,...,pn)

Basic properties of determinants These properties follow immediately from the definition. On the other hand, they characterize the determinant. Only det(A) satisfies them.

  1. Alternating function. Interchanging two rows changes the sign of the determinant. Thus, if E is a type I elementary matrix that interchanges two rows, then det(EA) = - det(A).
  2. Homogeneity. If Ea is a type II elementary matrix that multiplies a fixed row by a, then det(EaA) = a det(A).
  3. Additivity. det A is an additive function of a fixed row. This means that if A, B, and C are identical except that rowi(A) = rowi(B) + rowi(C), then det(A) = det(B) + det(C).
  4. det(I) = 1, I = identity matrix.

Determinants and matrices
  1. If two rows of A are equal, then det(A)=0.
  2. If a row of A has all 0's, then det(A)=0.
  3. Product rule
    If A and B are n×n matrices, then det(AB)=det(A)det(B).
  4. A is singular if and only if det(A) = 0.
  5. det(A-1) = (det(A) )-1
  6. If Ec is a type III row operation that replaces rowi by rowi + c rowj, then det (EcA) = det(A).
  7. det(AT) = det(A), so all of the statements above apply to columns as well as rows.

Cofactor expansions Consider the matrix A with rowi replaced by ej = (0 0 ... 1 ...0), where the 1 is in position j. We call the determinant of this new matrix the (i,j) cofactor, and we denote it by Aij. This cofactor is related the (i,j) minor, Mij, which is the determinant of the (n-1)×(n-1) matrix formed by removing row i and column j from A, via
Aij = (-1)i+jMij.
The deteminant of A can be found via expnsion about row i:
det(A) = ΣjaijAij
Similarly, one may use a column expansion:
det(A) = ΣiaijAij
These are nearly identical. The difference between them is that the summation index is different in the two formulas, the first being over the column index j, and the second over the row index i.