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Determinants and Cramer’s Rule

Determinants and Cramer’s Rule. Determinants. Determinants. A real number A ssociated with square (n x n) matrices Determinant for Matrix A is denoted as det A or | A |. Finding the Determinant of a 2 x 2 Matrix:. Example 1: Find the determinant. Example 2: Find the determinant.

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Determinants and Cramer’s Rule

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  1. Determinants and Cramer’s Rule

  2. Determinants

  3. Determinants • A real number • Associated with square (n x n) matrices • Determinant for Matrix A is denoted as det A or | A |

  4. Finding the Determinant of a 2 x 2 Matrix:

  5. Example 1: Find the determinant

  6. Example 2: Find the determinant

  7. Example 3: Find the determinant

  8. Practice 1: Find the determinant

  9. Cramer’s Rule

  10. Cramer’s Rule • Method for solving linear systems using matrices -Can be used to solve a linear system of any size *We will only be working with linear systems in two variables

  11. How to Solve Using Cramer’s Rule: • Create a coefficient matrix from the linear system ax + by = e cx + dy = f

  12. How to Solve Using Cramer’s Rule: • Create a coefficient matrix from the linear system ax + by = e cx + dy = f *Note that the coefficients for x are the first column and the coefficients for y are the second column

  13. How to Solve Using Cramer’s Rule: • Find the determinant of the coefficient matrix

  14. How to Solve Using Cramer’s Rule: • Find the determinant of the coefficient matrix *If det A ≠ 0, then there is exactly one solution

  15. How to Solve Using Cramer’s Rule: 3. Solve for x as follows: ax + by = e cx + dy = f

  16. How to Solve Using Cramer’s Rule: 3. Solve for x as follows: ax + by = e cx + dy = f *Note the coefficients for x have been replaced with the constants from the linear system

  17. How to Solve Using Cramer’s Rule: 4. Solve for y as follows: ax + by = e cx + dy = f

  18. How to Solve Using Cramer’s Rule: 4. Solve for y as follows: ax + by = e cx + dy = f *Note the coefficients for y have been replaced with the constants from the linear system

  19. How to Solve Using Cramer’s Rule: • Write the solution as a point (x,y)

  20. How to Solve Using Cramer’s Rule: • Write the solution as a point (x,y) 6. Check your solution by substituting the values into the original equation

  21. Example 1: Solve the following system using Cramer’s Rule: 2x + 7y = –3 3x – 8y = –23

  22. Example 2: Solve the following system using Cramer’s Rule: 4x + 12y = 11 14x – 8y = 1

  23. Practice 1: Solve the following system using Cramer’s Rule: 3x + 19y = 20 2x + 11y = 10

  24. Homework: Page 607 #1 – 15

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