University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A23
Winter 2008
Lecture 17
Section 4.1,
Determinants
Sophie Chrysostomou
Areas, Volumes and Cross Products
definition
0.1
(i) The
determinant of
1
×
1
matrix
is its sole entry.
(ii)The
determinant of a
2
×
2
matrix
is given by
det
(
A
) =
a
1
a
2
b
1
b
2
=
a
1
b
2

b
1
a
2
,
where
A
=
a
1
a
2
b
1
b
2
(iii) The
determinant of a
3
×
3
matrix
is given by
det
(
A
) =
a
1
a
2
a
3
b
1
b
2
b
3
c
1
c
2
c
3
=
a
1
b
2
b
3
c
2
c
3

a
2
b
1
b
3
c
1
c
3
+
a
3
b
1
b
2
c
1
c
2
definition
0.2
If
a
= [
a
1
, a
2
, a
3
]
,
b
= [
b
1
, b
2
, b
3
]
∈
R
3
the
cross product
of
a
and
b
is given by
a
×
b
=
a
2
a
3
b
2
b
3
,

a
1
a
3
b
1
b
3
,
a
1
a
2
b
1
b
2
=
i
a
2
a
3
b
2
b
3

j
a
1
a
3
b
1
b
3
+
k
a
1
a
2
b
1
b
2
Note:
a
×
b
is perpendicular to both
a
and
b
. (Show this on your own.)
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theorem
0.3
(i) If a parallelogram is determined by two nonzero vectors,
a
= [
a
1
, a
2
]
and
b
= [
b
1
, b
2
]
in
R
2
, then its area is given by
Area
=

a
1
b
2

a
2
b
1

=
det
a
1
a
2
b
1
b
2
=
a
1
a
2
b
1
b
2
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 Spring '10
 Sophie
 Linear Algebra, Algebra, Determinant, Characteristic polynomial, Howard Staunton

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