3.3 Determinants & Cramer’s Rule

1. 3.3 Determinants & Cramer’s Rule Algebra II Mrs. Spitz Fall 2006

2. Objectives: • Determine the determinants of a 2x2 and 3x3 matrix • User Cramer’s rule to solve systems of linear equations.

3. Assignment • Pp.120-121 #4-24 all;25-39 odd

4. Determinants Associated with every square matrix is a whole number called the determinant The Determinant of a Matrix A is denoted by detA or |A|

5. Determinant of a 2x2 = ad - cb

6. Ex 8 Evaluate the determinant =1(7) — 2(4) = 7 - 8 = -1

7. Ex 9 Evaluate the determinant =7(3) — 2(2) = 21 - 4 = 17

8. Ex 10 Evaluate • -3 • -2 0 • 1 2 (2*0*6 + 4*-2*2) = -3*3*1 + (1*0*4 + 6*-2*-3) ---- + 2*3*2 Step 1: recopy the first two columns. = (0 + -9 + -16) – (0 + 12 + 36) Step 2: multiply the down diagonals and add the products. = -25 - 48 Step 3: multiply the up diagonals and add the products NOTE: You subtract the up diagonal from the down diagonal = -73

9. Evaluate. Ex 11 det = -89

10. Using the TI-83 • Go to the Matrix menu • Go to Edit • Enter the dimensions • Enter the values into the matrix

11. Determinant of a 3x3 From the calculator!! • Enter the given matrix into matrix A. • Go to the home screen • Go to the Matrix menu • Go to Math • Choose 1: det • Select matrix A. • Press Enter

12. Cramer’s Rule (because Cramer RULES!) Gabriel Cramer was a Swiss mathematician (1704-1752)

13. Coefficient Matrices • You can use determinants to solve a system of linear equations. • You use the coefficient matrix of the linear system. • Linear SystemCoeff Matrix ax+by=e cx+dy=f

14. Cramer’s Rule for 2x2 System • Let A be the coefficient matrix • Linear SystemCoeff Matrix ax+by=e cx+dy=f • If detA 0, then the system has exactly one solution: and

15. Example 1- Cramer’s Rule 2x2 • Solve the system: • 8x+5y=2 • 2x-4y=-10 The coefficient matrix is: and So: and

16. Solution: (-1,2)

17. Example 2- Cramer’s Rule 2x2 • Solve the system: • 2x+y=1 • 3x-2y=-23 The solution is: (-3,7) !!!