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18.02 Multivariable Calculus
Fall
200
7
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det(A),
1A.J.

it.'
a e i + b f g + d h c  c e g  b d i  f h a .
I
I
IAl
D1.
IAl
IAl
0
IAl
c,
IAl
(I),
+
1
Laplace
(1).
here
D. Determinants
Given a square array A of numbers, we associate with it a number called the
determinant
of A, and written either
or
For 2
x
2 and
3
x
3
arrays, the number is defined by
a
b
c
(1)
=
ad

bc;
=
d
e
f
g
h
i
Do not memorize these as formulas
learn instead the patterns which give the terms. The
2
x
2
case is easy: the product of the elements on one diagonal (the
"
main diagonal
"
), minus
the product of the elements on the other (the
"
antidiagonal
"
).
For the
3
x
3
case, three products get the
+
sign: those formed from the main diagonal,
or having two factors parallel to the main diagonal. The other three get a negative sign:
those from the antidiagonal, or having two factors parallel to
Try the following example
on your own, then check your work against the solution.
1

2
1
Example
1.1
Evaluate

1
3
2 using
2

1
4
Solution.
Using the same order as in
we get 12
(

8)
+

6

8

(

2)
=

7
.
Important facts about
:
is multiplied by

1
if we interchange two rows or two columns.
D

2.
=
if one row or column is all zero, or if two rows or two columns are the
same.
D

3.
is multiplied by
if every element of some row or column is multiplied by c.
of another row (resp. column).
D

4.
The value of
is unchanged if we add to one row (or column) a constant multiple
All of these facts are easy to check for 2
x
2 determinants from the formula (1); from this,
their truth also for
3
x
3
determinants will follow from the
expansion.
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 Spring '08
 Auroux
 Determinant, Addition, Multivariable Calculus, Laplace, Laplace expansion

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