INTRODUCTION to DETERMINANTS
 det(A):
 For a square matrix A
 If n = 1,
det(A) = a_{11}.
 If n = 2,
det(A) = a_{11}a_{22}a_{21}a_{12}.
 For ,
let
A_{ij} be A with row i and col j removed;
cofactor expansion for det A using first row of A.
 The
(i,j)^{th} cofactor of A is
C_{ij}=()^{i+j}det(A_{ij}).
 det(A) can be computed using a cofactor expansion along any row or
any column.
 Notation:
 Easy Determinants

det(I) = 1.
 det for a triangular A is the product of the diagonal entries.
PROPERTIES OF DETERMINANTS
 Row Operations:
 A row interchange for A changes then sign of det(A).
 The scale of a row of A by s changes det(A) to s(det(A)).
 Row replacement operations do not change det(A).
 If
(upper triangular) and r was the number of
row interchanges
,
 A square matrix A is invertible iff
.
 Other Properties:
 det(A) = det(A^{T}).
 det(AB) =
det(A)det(B)
simple consequences:

det(A^{1}) = (det(A))^{1}

det(A^{n})=(det(A))^{n}

det(sA) = s^{n} det(A)

det(AB) = det(BA)
DETERMINANT FORMULAS
 Cramer's Rule:
 Let
denote A with col i replaced by .
The solution to
is given by
,
for
.
 A^{1} Formula:
 Using adj(A) to denote the adjugate or adjoint of A
(adj(A) is the transposed matrix of cofactors C_{ij} for A),
For n=2,
.
 Areas and Volumes:

 A parallelogram determined by vectors ,
has
area
.
 A parallelepiped determined by vectors ,
,
has
volume
.
Alan C Genz
20000706