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History. Charles Babbage (1792-1871) knew of Cramer’s Rule from early 18 th century mathematician Gabriel Cramer. Cramer’s rule was simple but involved numerous multiplications for large systems. - PowerPoint PPT Presentation
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History
1. Charles Babbage (1792-1871) knew of Cramer’s Rule from early 18th century mathematician Gabriel Cramer. Cramer’s rule was simple but involved numerous multiplications for large systems.
2. Babbage designed a machine, called the “difference engine” for performing these operations. His invention demonstrated how complex calculations could be handled mechanically. In 1944, IBM used the lessons of his difference engine to create the world’s first computer.
10.3 Systems of Linear Equations; Determinant
1. Evaluate 2 x 2 Determinants
Definition: Determinant of a 2 x 2 Matrix is the value
Notation: represents the determinant (a single value)
represents the matrix
dc
ba
dc
ba
bcaddc
baD
1 Evaluate 2 x 2 Determinants
Examples: Evaluate the following :
1)
2)
3)
37
65
53
42
52
104
2. Cramer’s Rule for a 2x2
Given the system:
Solution is:
where:
If this method can not be used.
D
Dx x
df
beDx
D
Dy y
dc
baD
fc
eaDy
fdycx
ebyax
0D
Evaluate the determinant of the 3x3 matrix:
3 a) Determinant of a 3 by 3 system
333
222
111
cba
cba
cba
333
222
111
cba
cba
cba
333
222
111
cba
cba
cba
Definition: The minor of an element is the determinant that remains after deleting the row and column of that element
33
221
33
221
33
221
333
222
111
ba
bac
ca
cab
cb
cba
cba
cba
cba
Practice
Examples: Evaluate the following :
1)
211
803
241
2 b) Example
Use Cramer’s Rule to solve the system:
156
245
yx
yx
P. 767 #16. Solve 3 ways!#21 Which method would you prefer for this problem ?#24. D=0.
2 c) Why does Cramer’s Rule work?
A solution for the system:
Proof:Step 1: Using elimination, add the 2 equations together to eliminate the
y variable.Step 2: Solve for xStep 3: Replace the numerator and denominator of x with the definition
of a determinant.Step 4: Repeat steps 1-3 for y.
fdycx
ebyax
How do we use Cramer’s Method for 3 x 3 systems?
3 Cramer’s Rule for solving a 3 x 3 system
3333
2222
1111
ezcybxa
ezcybxa
ezcybxa
Use the notation for minors to write the determinant:
3 c) Cramer’s Rule for 3 by 3 system
333
222
111
cba
cba
cba
D
333
222
111
cbe
cbe
cbe
Dx
333
222
111
cea
cea
cea
Dy
333
222
111
eba
eba
eba
Dz
D
Dx x
D
Dy y
D
Dz z
P. 767 #34
#33
When D = 0, the system is either inconsistent or dependent.
Two cases…Inconsistent: when D=0 and at least one of the determinants in the
numerator is not 0. example:
Dependent: when D=0 and all numerators are 0. example:
4. Special Cases – Cramer’s Rule does not apply
3 ,0 ,0 yx DDD
0 ,0 ,0 yx DDD
