MA 3280 Lecture 19  Determinants
Monday, March 24, 2014.
Objectives: Material from 8.2 and 8.3: Order 1, 2, and 3 determinants.
At the beginning of the semester, and when we talked about Fnding inverse matrices, we used something
that is called a
determinant
. ±or example, given a 2
×
2 matrix
(1)
A
=
b
a
11
a
12
a
21
a
22
B
,
we saw that
(2)
A

1
=
1
a
11
a
22

a
12
a
21
b
a
22

a
12

a
21
a
11
B
That funny thing on the bottom is the determinant of
A
. We’ll use vertical line brackets to denote the
determinant, like

A

. A 2
×
2 determinant, therefore, is deFned to be
(3)
v
v
v
v
a
11
a
12
a
21
a
22
v
v
v
v
=
a
11
a
22

a
12
a
21
.
As an example,
(4)
v
v
v
v
3
4
7
2
v
v
v
v
= (3)(2)

(4)(7) = 6

28 =

22
.
Cramer’s Rule.
The little trick I showed you to solve a 2
×
2 system of equations can be reformulated in
terms of determinants. It turns out that the solution for a system like
(5)
2
x
1
+
3
x
2
=
1
5
x
1
+
4
x
2
=
6
has solutions as follows.
(6)
x
1
=
v
v
v
v
1
3
6
4
v
v
v
v
v
v
v
v
2
3
5
4
v
v
v
v
=
4

18
8

15
=

14

7
= 2
,
and
(7)
x
2
=
v
v
v
v
2
1
5
6
v
v
v
v
v
v
v
v
2
3
5
4
v
v
v
v
=
12

5
8

15
=
7

7
=

1
.
Do you see the pattern? The determinant on the bottom has the coe²cients on the left side of the equations,
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 Spring '14
 Linear Algebra, Algebra, Determinant, Matrices, Parity, Chess opening, Howard Staunton, Cramer, transpositions

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