MATH 17100
28446
11/16/2009
Determinants
The
determinant
of a
square matrix
A is a scalar denoted by
det
or
A
A
obtained as
follows:
The determinant of the
1 1
×
matrix
11
[
]
a
=
A
is
11
11
det
det[
]
a
a
=
A
.
The determinant of the
2
2
×
matrix
11
12
21
22
a
a
a
a
=
A
is
11
12
11
22
12
21
21
22
det
det
a
a
a a
a a
a
a
=
=
=
A
.
The determinants of higher order matrices are obtained by any of the following methods:
(I)
Method of Laplace Expansion
Corresponding to the element
ij
a
of the square matrix
A
, there is a
submatrix
obtained by
eliminating the row and column occupied by the element. Let us denote that submatrix by
ij
S
.
The determinant of this submatrix is called the (
i
,
j
)
minor
of
A
and denoted
det
ij
ij
M
=
S
. The
(
i
,
j
)
cofactor
of
A
is defined as
( 1)
i
j
ij
ij
A
M
+
=
−
.
The determinant of an
n
n
×
matrix
A
is obtained by a
cofactor expansion
along any row or
column of
A
:
1
det
n
ij
ij
j
a A
=
=
∑
A
, cofactor expansion along the
th
i
row
1
det
n
ij
ij
i
a A
=
=
∑
A
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 Spring '10
 KITCHENS
 Determinant, Scalar, Characteristic polynomial, Triangular matrix, Det

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