(Section 8.5: Applications of Determinants)
8.67
SECTION 8.5: APPLICATIONS OF DETERMINANTS
PART A: CRAMER’S RULE FOR SOLVING SYSTEMS
A square system of linear equations is a system of
n
linear equations in
n
unknowns,
where
n
∈
Z
+
. Cramer’s Rule uses determinants to solve such a system. For now, we
assume that the unknowns are
x
,
y
, etc. and that they make up
X
, the vector of
unknowns.
Cramer’s Rule
Write the augmented matrix for the system
AX
=
B
:
A
B
⎡
⎣
⎤
⎦
•
A
is the coefficient matrix.
If the system is square,
A
will be a square matrix.
•
B
is the righthand side (
RHS
); you could use
RHS
, instead.
Compute the following determinants:
• Let
D
=
A
, or det
A
( )
.
• Let
D
x
=
A
x
, or det
A
x
( )
.
where
A
x
is identical to
A
, except that the
x
column of
A
is
replaced by
B
, the
RHS
.
(continued on next page)
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8.68
Cramer’s Rule (cont.)
• Let
D
y
=
A
y
, or det
A
y
( )
,
where
A
y
is identical to
A
, except that the
y
column of
A
is
replaced by
B
, the
RHS
.
•
D
z
,
A
z
, etc. are defined analogously as necessary.
If
D
≠
0
,
there is a unique solution
given by:
x
=
D
x
D
,
y
=
D
y
D
,
z
=
D
z
D
(if applicable), etc.
If
D
=
0
, there is
not
a unique solution. Then:
• If all of the other determinants,
D
x
,
D
y
, etc. are also 0, then the
system has infinitely many solutions.
• Otherwise, the system has no solution. The solution set is
∅
,
the empty set.
Note: Observe that the formulas for
x
,
y
, etc. fall apart if
D
=
0
.
Note: In fact, if
A
is square, then its determinant
D
≠
0
if and only if
A
is invertible,
which is true if and only if
AX
=
B
has a unique solution (given by
X
=
A
−
1
B
). See the
Inverse Matrix Method for solving systems in
Section 8.3, Part D
.
Note: One advantage that this method has over Gaussian Elimination with Back
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 Fall '11
 staff
 Calculus, Linear Algebra, Determinant, Linear Equations, Equations, Vector Space, Invertible matrix, Det Ax

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