Determinants General

Definition Let A =
2
4
a11 a12 a13
a21 a22 a23
a31 a32 a33
3
5. Then the determinant of A is the
scalar
detA = jAj
= a11
¯¯¯¯
a22 a23
a32 a33
¯¯¯¯
? a12
¯¯¯¯
a21 a23
a31 a33
¯¯¯¯
+ a13
¯¯¯¯
a21 a22
a31 a32
¯¯¯¯
detA = a11 detA11 ? a12 detA12 + a13 detA13
=
X3
j=1
(?1)1+ja1j detA1j
For any n£n matrix A, the determinant detAij of the (n?1)£(n?1) matrix
obtained from A by deleting the i-th row and the j-th column is called the
(i; j)-minor of A.
Determinants of n £ n Matrices
Definition Let A = [aij ] be an n £ n matrix, where n ¸ 2. Then the
determinant of A is the scalar
detA = jAj
= a11 detA11 ? a12 detA12 + ¢ ¢ ¢ + (?1)1+na1n detA1n
=
Xn
j=1
(?1)1+ja1j detA1j
It is convenient to combine a minor with its plus or minus sign. To this end,
we define the (i, j)-cofactor of A to be
Cij = (?1)i+j detAij
With this notation, the definition becomes
detA =
Xn
j=1
a1jC1j
One of the exercises in the textbook asks you to check that this definition
correctly gives the formula for the determinant of a 2£2 matrix when n = 2.
This definition is often referred to as cofactor expansion along the first
row.
MATH10212 ² Linear Algebra ² Brief lecture notes 40
Theorem 4.1. The Laplace Expansion Theorem
The determinant of an n£n matrix A = [aij ], where n ¸ 2, can be computed
as
detA = ai1Ci1 + ai2Ci2 + ¢ ¢ ¢ + ainC1n
=
Xn
j=1
aijCij
(which is the cofactor expansion along the ith row) and also as
detA = a1jC1j + a2jC2j + ¢ ¢ ¢ + anjCnj
=
Xn
i=1
aijCij
(the cofactor expansion along the jth column).
Since
Cij = (?1)i+j detAij ;
each cofactor is plus or minus the corresponding minor, with the correct
sign given by the term (?1)i+j . A quick way to determine whether the sign
is + or ? is to remember that the signs form a “checkerboard" pattern:
2
666664
+ ? + ? ¢ ¢ ¢
? + ? + ¢ ¢ ¢
+ ? + ? ¢ ¢ ¢
? + ? + ¢ ¢ ¢
...
...
...
...
. . .
3
777775
Theorem 4.2. The determinant of a triangular matrix is the product of
the entries on its main diagonal. Specifically, if A = [aij ] is an n £ n triangular
matrix then
detA = a11a22 ¢ ¢ ¢ ann
Properties of Determinants
The most efficient way to compute determinants is to use row reduction.
However, not every elementary row operation leaves the determinant of a
matrix unchanged. The next theorem summarizes the main properties you
need to understand in order to use row reduction effectively.
Theorem 4.3. Let A = [aij ] be a square matrix.
a. If