Transcript coefficient matrix
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.
det
a
c b d
a c b d
ad
cb
Determinant of a 3
3 matrix
A
3
3
matrix has a different consideration, aside from just its diagonals.
det
a
d g
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.
b e h c f i
a d g b e h f c i a d g b e h
aei
bfg
cdh
gec
hfa
idb
Example 1
Evaluate the determinant of each matrix.
6 1 2 4 4 1 2 2 1 5 0 2 3
Area of a triangle
The area of a triangle whose vertices are at (
x
1 ,
y
1 ), (
x
2 ,
y
2 ), and (
x
3 ,
y
3 ) is given by the formula:
x
1
y
1 1
(x 1 , y 1 )
Area 1 2
x
2
x
3
y
2
y
3 1 1
(x 2 , y 2 ) (x
, y 3 )
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 Coefficient Matrix
ax
cx
by
dy
e
f
a
c
x
coefficients
b d
y
coefficients
Cramer’s Rule Steps
Let
A
below.
be the coefficient matrix of the system If det
A
of equations has exactly one solution.
≠ 0, then the system
ax cx
by dy
e
The solution is:
e b a e f x
f
det
d A y
c
det
f 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: 6
x
3
x
2
y
5
y
16 16
Cramer’s Rule for 3
3 Systems
Let
A
be the coefficient matrix for the system of equations shown.
ax
dx gx
by
ey
hy
If det
A
≠0, the system has exactly one solution. The solution is:
k j b e c f a d k j
f a d b e
cz
iz
k j fz
l k j
l h i g l i g h l x
y
z
det
A
det
A
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).
Compound Formula Atomic weight
Sodium fluoride FNa 42 Sodium chloride Chlorine pentafluoride NaCl ClF 5 58.5
130.5
Assessment
Worksheet 3.7B