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18.02 Multivariable Calculus
Fall 2007
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1A.J.

it.'
aei+bfg+dhccegbdifha.
I
I
IAl
D1.
0
c,
(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
abc
(1)
=
ad

bc;
=
d
e
f
ghi
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.
D2.
=
if one row or column is all zero, or if two rows or two columns are the
same.
D3.
is multiplied by
if every element of some row or column is multiplied by c.
of another row (resp. column).
D4.
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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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.
 Spring '08
 Auroux
 Determinant, Multivariable Calculus

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