MATH 1314 College Algebra Notes
Spring 2013
Chapter 8.3:
Determinants and Cramer’s Rule
A determinant is a rectangular array of numbers. Cramer’s Rule will be used
to solve Linear Systems. Cramer’s Rule uses ratios of determinants from a
Linear System to find a solution(if one exists). If a Linear System has no
solution or infinitely many solutions, then Cramer’s Rule does not apply..
 Evaluate a Second-Order Determinant; Solve Linear System in two variables
 A determinant is a real number associated with a square matrix having
the same number of rows and columns.
The determinant of matrix [
] is written |
|.
The value of this second-order determinant is given by the cross
product formula ( )( ) ( )( ).
The determinant of matrix [
] is written |
|.
The value of this third-order determinant can be evaluated with a
method called expansion by minors.
A formula for expanding |
(
)
|
|
(
| by minors along first row
)
|
|
(
)
|
is
|.
In this expansion, each term is made of 3 factors: ( ) , entry from ,
and a 2x2 determinant. For the first term, the exponent in ( )
comes
from the row( ) and column( ) corresponding to entry . The
corresponding 2x2 determinant(or minor) for entry
is the result of
crossing out entries in
and
in the main determinant. The product of
( )
and its minor is called the cofactor of entry . Look for a row or
column containing zeros to make expanding by minors easier.
 To evaluate a determinant for a matrix on the TI-83/84 calculator:
 Press 2nd
for MATRX menu and select EDIT menu.
Select matrix [A] and enter number of rows and columns, then
matrix coefficients. Press 2nd QUIT
 Press 2nd
for MATRX menu. Under MATH menu, select
the command det( and press Enter to bring command to screen.
 Press 2nd
for MATRX menu. Under NAMES menu,
select matrix [A] and press Enter. Press Enter again to evaluate.
MATH 1314 College Algebra Notes
Chapter 8.3:
Spring 2013
Determinants and Cramer’s Rule
 To solve a Linear System in two variables using Cramer’s Rule will
require:
 Forming and evaluating the determinants ,
, and
 then finding solutions for and with the values of ,
, and
In general, for a Linear System in two variables{
the determinants are
|
and the solution is
and
|,
,
|,
|
, provided that
.
|
|
Important: Whenever determinant
, the Linear System cannot be
solved with Cramer’s Rule.
If
and
or
, then the Linear System is
__inconsistent__ and the system has __no solution__.
If
,
, and
, then the equations in the Linear System
are _dependent__ and the system has __infinitely many solutions__.
 To solve a Linear System with determinants on the TI-83/84
calculator:
 Press 2nd
for MATRX menu and select EDIT menu.
Select matrix [A] and enter number of rows and columns, then
coefficients for
. Press 2nd QUIT
Repeat with matrix [B] for
.
Repeat with matrix [C] for
and so on as needed.
nd
 Press 2
for MATRX menu. Under MATH menu,
select determinant command det( . Press ENTER.
Press 2nd
for MATRX menu again. Under NAMES
menu, select matrix [A]. Press ENTER to bring matrix [A] to
screen. Press ENTER again to evaluate det( [A] ).
Repeat to evaluate det( [B] ) and det( [C] ) .
 Solutions will be
,
, and so on, if they exist.
Cramer’s Rule
Practice problems
Step 1: Set up determinants
|
|,
,
|
,
Spring 2013
.
|,
|
|
Step 2: Use cross product formula for second-order determinants or TI-83/84 det(
command to evaluate each square determinant.
(If
, then the system either has infinitely many solutions or no solution,
therefore Cramer’s Rule is not applicable in determining a solution set.)
,
Step 3: Since
,
, Cramer’s Rule is not applicable in determining a solution set.
Select Answer Choice B.
Cramer’s Rule
Practice problems
Step 1: Set up determinants
|
|,
|
,
Spring 2013
,
,
.
|
|,
|
|,
|
Step 2: Use Expansion by Minors method for third-order determinants or TI-83/84
det( command to evaluate each square determinant.
(If
, then the system either has infinitely many solutions or no solution,
therefore Cramer’s Rule is not applicable in determining a solution set.)
,
Step 3: Since
,
,
is not zero, the system has solutions and are given by
,
,
Select Answer Choice A and enter solutions