Transcript coefficient matrix
Evaluate Determinants and Apply Cramer’s Rule
Algebra 2 Mr. Keltner
Determinants of a square matrix
, there is a real number associated with it called its
The determinant of matrix
is written as: det
The determinant of a 2 2 matrix is the difference of the products of its diagonals.
c b d
a c b d
Determinant of a 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.
b e h c f i
a d g b e h f c i a d g b e h
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 (
1 ), (
2 ), and (
3 ) is given by the formula:
(x 1 , y 1 )
Area 1 2
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
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).
We can use determinants to solve a system of linear equations, using a method called
, using the
of the linear system.
The coefficient matrix simply aligns the coefficients of the variables in the system of linear equations.
Linear System Coefficient Matrix
Cramer’s Rule Steps
be the coefficient matrix of the system If det
of equations has exactly one solution.
≠ 0, then the system
The solution is:
e b a e f x
d A y
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
Cramer’s Rule for 3
be the coefficient matrix for the system of equations shown.
≠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
k j fz
l k j
l h i g l i g h l x
Notice, again, the variable’s coefficients are replaced by the column of constants in each numerator.
Example 4: 3
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