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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.

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History

History

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.

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.


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10.3 Systems of Linear Equations; Determinant


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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


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1 Evaluate 2 x 2 Determinants

Examples: Evaluate the following :

1)

2)

3)


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2. Cramer’s Rule for a 2x2

Given the system:

Solution is:

where:

If this method can not be used.


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3 a) Determinant of a 3 by 3 system

Evaluate the determinant of the 3x3 matrix:

Definition: The minor of an element is the determinant that remains after deleting the row and column of that element


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Practice

Examples: Evaluate the following :

1)


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2 b) Example

Use Cramer’s Rule to solve the system:

P. 767 #16. Solve 3 ways!

#21 Which method would you prefer for this problem ?

#24. D=0.


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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 x

Step 3: Replace the numerator and denominator of x with the definition of a determinant.

Step 4: Repeat steps 1-3 for y.


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3 Cramer’s Rule for solving a 3 x 3 system

How do we use Cramer’s Method for 3 x 3 systems?


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3 c) Cramer’s Rule for 3 by 3 system

Use the notation for minors to write the determinant:

P. 767 #34


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#33


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4. Special Cases – Cramer’s Rule does not apply

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:


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