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.
- 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).
- Homogeneity. If Ea is a type II elementary matrix
that multiplies a fixed row by a, then det(EaA) = a
det(A).
- 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).
- det(I) = 1, I = identity matrix.
Determinants and matrices
- If two rows of A are equal, then det(A)=0.
- If a row of A has all 0's, then det(A)=0.
- Product rule
If A and B are n×n matrices, then det(AB)=det(A)det(B).
- A is singular if and only if det(A) = 0.
- det(A-1) = (det(A) )-1
- If Ec is a type III row operation that replaces
rowi by rowi + c rowj, then det
(EcA) = det(A).
- 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.