12.3
Systems of Linear Equations:
Determinants
One method for solving systems of linear equations is called
Cramer’s Rule
and is based on
determinants.
This method can only be used when the number of equations equals the number
of variables.
Evaluating 2
×
2 Determinants:
If
a, b, c,
and
d
are real numbers, then
a
b
D
ad
bc
c
d
=
=

.
Evaluate:
1.
3
2
5
8

2.
3
5
2
4


Solving a system of two equations in two variables:
Given the system
ax
by
s
cx
dy
t
+
=
+
=
,
x
s
b
s
b
t
d
t
d
D
x
a
b
D
D
c
d
=
=
=
and
y
a
s
a
s
D
c
t
c
t
y
a
b
D
D
c
d
=
=
=
, D ≠ 0
If D = 0, Cramer’s Rule cannot be used; the system is either inconsistent of dependent.
Solve:
3.
5
3
21
4
7
2
x
y
x
y

=
+
= 
4.
8
2
3.4
6
3.3
x
y
x
y

=


= 
Evaluating 3
×
3 Determinants:
A 3
×
3 determinant is symbolized by
11
13
2
12
21
2
2
31
3
32
3
3
a
a
a
a
a
a
a
a
a
, where the
double subscript indicates its row and column; e.g., entry a
32
is in row 3 column 2.
The value of a 3
×
3
determinant is defined in terms of 2
×
2 determinants, called minors of the 3
×
3 determinant.
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 Fall '08
 Staff
 Determinant, Linear Equations, Equations, Systems Of Linear Equations, Howard Staunton, Cramer, a12, a32, a13

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