1. DETERMINANTS SECTION 6.3

2. DETERMINANTS 2 X 2 Matrices:Det A = ad - bc

3. SIGN CONVENTION We associate a plus or minus sign with every position in a matrix. 3 X 3 Matrices:

4. SIGN CONVENTION We associate a plus or minus sign with every position in a matrix. 4 X 4 Matrices:

5. EXAMPLE:

8. - 3 [ 8 - 6] - 2 [ 2 + 10] + 3 [ - 3 - 20] - 6 - 24- 69 - 99

9. DETERMINANTS ON CALCULATOR DET A = 154

10. EXAMPLE: Solve for x:

11. EXAMPLE: x 2 (2 - 6) - x (1 - 12) + 1 (2 - 8) = 0 - 4 x 2 + 11x - 6 = 0 x = 3/4x = 2

12. CRAMER’S RULE For the system:ax + by = s cx + dy = t D = D x =D y = The solution is: x = y =

13. AN IMPORTANT CONCLUSION: If the determinant of the coefficient matrix is 0, the system has no unique solution. If the determinant of the coefficient matrix is 0, the system has no unique solution. This means that the system is either inconsistent (has no solution) or it is dependent (has infinitely many solutions. This means that the system is either inconsistent (has no solution) or it is dependent (has infinitely many solutions.

14. CRAMER’S RULE a 11 x + a 12 y + a 13 z = c 1 a 21 x + a 22 y + a 23 z = c 2 a 31 x + a 32 y + a 33 z = c 3

15. CRAMER’S RULE

16. The solution is: x =y =z =

17. EXAMPLES Do Problems 34, 36, and 38.

18. CONCLUSION OF SECTION 6.3 CONCLUSION OF SECTION 6.3