1 / 13

History

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.

cheryl-vazquez
Download Presentation

History

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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.

  2. 10.3 Systems of Linear Equations; Determinant

  3. 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

  4. 1 Evaluate 2 x 2 Determinants Examples: Evaluate the following : 1) 2) 3)

  5. 2. Cramer’s Rule for a 2x2 Given the system: Solution is: where: If this method can not be used.

  6. 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

  7. Practice Examples: Evaluate the following : 1)

  8. 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.

  9. 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.

  10. 3 Cramer’s Rule for solving a 3 x 3 system How do we use Cramer’s Method for 3 x 3 systems?

  11. 3 c) Cramer’s Rule for 3 by 3 system Use the notation for minors to write the determinant: P. 767 #34

  12. #33

  13. 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:

More Related