#### 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