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Friday March 13
−
Lecture 27 :
Determinants
(Refers to 5.1)
Expectations
:
1.
Define the (
i
,
j
)
minor
of a square matrix.
2.
Define the (
i
,
j
)
cofactor
of a square matrix.
3.
Calculate the determinant of a square matrix by row or column expansion.
4.
Apply the fact that the determinant of a triangular matrix is the product of the
entries on the diagonal.
5.
Recognize that a
triangular matrix A is invertible iff its determinant is not
0.
6.
Recognize the effects the 3 elementary row (column) operations have on the value
of the determinant of a matrix.
7.
Recognize that det(
A
) = det(
A
T
).
8.
Recognize conditions on
A
that force det(
A
) = 0.
9.
Find determinants by using the properties of determinants.
27.1
Definition
−
A
determinant
is a function
D
which maps a square matrix
A
to a real
number according to well defined formula applied to the entries of the matrix. We denote
this function by
det(
A
)
or

A

. In the case of a 2
×
2 matrix
A
= [
a
ij
]
2
×
2
,
det(
A
) =
a
11
a
22
−
a
12
a
21
.
The general definition for the determinant is of an
n
×
n
matrix is given after we have
developed some terminology.
27.2
Definition  Given a square
n
by
n
matrix
A
, for the entry
a
ij
in
A
, we associate a
matrix
M
ij
of dimension
n
−
1 by
n
−
1 whose entries are the ones that remain when we
remove from
A
the row and the column containing the entry
a
ij
. We call the number
m
ij
=
det(
M
ij
),
the minor of the element
a
ij
, or the
(
i, j
) minor
.
27.2.1
Remark
−
For an
n
×
n
matrix
A
there are
n
2
minors
m
ij
, one for each entry of
A
.
27.3
Definition  For each minor
m
ij
, we define the
cofactor
c
ij
as
c
ij
= (
−
1)
i + j
m
ij
.
27.3.1
Examples are given.
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View Full Document27.4
Definition  For an
n
×
n
matrix
A =
[
a
ij
]
n
x
n
, the determinant of
A
, det(
A
) = 
A
 is
defined as :
Det(
A
)
=
∑
j
= 1 to
n
a
1j
c
1j
=
a
11
c
11
+
a
12
c
12
+ .
.. +
a
1
n
c
1
n
.
27.4.1
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 Winter '08
 All
 Determinant

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