Download 4.3 Determinants and Cramer`s Rule

4.3 Determinants and Cramer’s
Rule
Determinants

The
the matrix
D

of

 a1
a
 2
a1
b1
a2
b2

b1 
b2 
, symbolized
has the value
a1b2  a2b1.
Note that the vertical bars are not absolute value
signs, it is notation for determinant.
Evaluate each determinant.
1.
8 2
3 7
2.
8 6
4
3
3.
k
3k
2k  1 3  6k
Given the system of equations with the
coefficient matrix:
a1 x  b1 y  c1

 a1
a
 2
a2 x  b2 y  c2
D
a1
b1
a2
b2
The determinant of
are the coefficient matrix
Dx 
c1
b1
c2
b2
The x coefficients are
replaced with the
constants c1 and c2

b1 
b2 
Dy 
a1
c1
a2
c2
The y coefficients
replaced with the
constants c1 and c2
Cramer’s Rule for a System:
Given the system of linear equations:
a1 x  b1 y  c1
a2 x  b2 y  c2
If the D  0, then the system has exactly one solution.
The solution is:
Dy
Dx
y
x
D
D

*Note: If D = 0, the system has either no solution or
infinitely many solutions.
Use Cramer’s Rule to solve the
system.
4.
2x  y  1
5.
3x  2y  23
4x  6y  4
x  5y  14
D=
Dx=
Dy=
D=
Dx=
Dy=
Point of Intersection:
Point of Intersection:
Use Cramer’s Rule to solve the
system.
6.
3x  4y  1
5x  7y  1
7. 7x  5y  0
5x  4 y  3
D=
Dx=
Dy=
D=
Dx=
Dy=
Point of Intersection:
Point of Intersection: