Evaluate Determinants and
Apply Cramer’s Rule
Section 3.7
Algebra 2
Mr. Keltner
Determinants of
a square matrix

For each square (n n) matrix, there is a real
number associated with it called its
determinant.

The determinant of matrix A is written as:



det A, or
|A|.
The determinant of a 2 2 matrix is the difference of
the products of its diagonals.
a b  a b
det 

 ad  cb

c d  c d
Determinant of a 3 3 matrix

A 3 3 matrix has a different
consideration, aside from just its
diagonals.
Copy & paste the first two columns to the
right of the determinant.
 Subtract the sum of one direction’s
diagonals from the sum of the other
direction’s diagonals.

a b
det d e


g h
c  a b
f  d e

i 
 g h
c a b
f d e  aei  bfg  cdh  gec  hfa  idb
i g h
Example 1

Evaluate the determinant of each matrix.
6 2 
1 4 


4 2 0 
1 1 2 



2 5 3 

Area of a triangle

The area of a triangle whose vertices are at
(x1, y1), (x2, y2), and (x3, y3) is given by the
formula:
x1 y1 1
1
(x1, y1)
Area  x 2 y 2 1
2
x3 y3 1

(x2, y2)
(x3, y3)


We use the ± symbol so that we can
choose the appropriate sign so that
our answer yields a positive value.
This is because we cannot have a
negative area.
Example 2: How big
is the city?

The approximate coordinates (in miles)
of a triangular region representing a city
and its suburbs are (10, 20), (-8, 5), and
(-4, -5).
Cramer’s Rule

Not this Kramer.
 We can use determinants to solve
a system of linear equations,
using a method called Cramer’s
rule, using the coefficient matrix
of the linear system.

The coefficient matrix simply
aligns the coefficients of the
variables in the system of linear
equations.
Linear System
ax  by  e

cx  dy  f
Coefficient Matrix
a b 
c d 


x
coefficients
y
coefficients
Cramer’s Rule Steps

Let A be the coefficient matrix of the system
below.
ax  by  e

 If det A ≠ 0, then the system

of equations has exactly one cx  dy  f
solution.
e b
a e
 The solution is:
f d
c f
x
y
det A
det A
 Notice the numerators are determinants that
replace the coefficients of each variable with
the column of constants.



Example 3: Cramer’s Rule

Use Cramer’s rule to solve the system:
6x  2y  16

3x  5y  16
Cramer’s Rule for
3  3 Systems
ax  by  cz  j
Let A be the coefficient matrix 

dx  ey  fz  k
for the system of equations
gx  hy  iz  l

shown.
 If det A≠0, the system has exactly one
solution. The solution is:

x


j
k
l
b c
e f
h i
det A
a j c

d k f
g l i
y
det A
a b j
d e k
g h l
z
det A
Notice, again, the variable’s coefficients are
replaced by the column of constants in each


numerator.
Example 4: 3 3 System


The atomic weights of three compounds are shown in
the table.
Use a linear system and Cramer’s rule to find the
atomic weights of fluorine (F), sodium (Na), and
chlorine (Cl).
Formula
Atomic
weight
Sodium fluoride
FNa
42
Sodium chloride
NaCl
58.5
Chlorine pentafluoride
ClF5
130.5
Compound
Assessment
Worksheet 3.7B