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